F: Exponentials and logarithms — AQA A-Level Mathematics
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F: Exponentials and logarithms explained
For a positive base a, a^x is defined for all real x and is always positive.
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Its graph passes through (0,1) and lies above the x-axis. When a>1 the graph increases and the x-axis (y=0) is a horizontal asymptote as x→-∞; when 0<a<1 it decreases and y=0 is an asymptote as x→+∞. The case a=1 is the constant function y=1, a horizontal line with no asymptote, so the asymptote statement applies only to a≠1. The base e≈2.718 is defined so that the gradient of e^x at x=0 is exactly 1, giving the self-derivative property. To sketch y=a^x, plot (0,1), (1,a) and (-1,1/a), then draw a smooth curve approaching but never touching the x-axis. Transformations: y=a^(x+b) shifts left by b, y=ka^x stretches vertically by k, and y=a^(-x) reflects in the y-axis.
Know that the gradient of e^(kx) is equal to k e^(kx) and hence understand why the exponential model is suitable in many applications.
Differentiating y = e^(kx) gives dy/dx = k e^(kx), so the gradient at any point equals k times the current value. The rate of change is therefore proportional to the quantity present, with constant of proportionality k: growth when k > 0, decay when k < 0. This self-replicating property makes e^(kx) suitable whenever the rate of change is proportional to the amount present, such as populations, radioactive decay and compound interest. For example, if N = N0 e^(0.05t), then dN/dt = 0.05N, so the rate is 5% of the current value per unit time. Newton's law of cooling also fits this model: defining y as the temperature difference from the surroundings gives dy/dt = -ky, an exponential decay. To use the result, differentiate term by term, apply the chain rule if needed, and interpret k as the continuous rate constant.
Know and use the definition of log_a(x) as the inverse of a^x, where a is positive and x ≥ 0. Know and use the function ln x and its graph. Know and use ln x as the inverse function of e^x.
This item requires you to treat logarithms as inverses of exponentials. For a positive base a (a ≠ 1), log_a(x) answers 'a to what power gives x?', so log_a(a^x)=x for all real x and a^{log_a(x)}=x for x>0. The domain of log_a(x) is x>0: log_a(0) and log_a(negative) are undefined in the reals, so the specification's 'x ≥ 0' should be read as x>0. The graph is the reflection of y=a^x in y=x, passing through (1,0) and (a,1), with a vertical asymptote at x=0. The natural logarithm ln x is the case a=e, so ln x is the inverse of e^x: ln(e^x)=x for all real x, and e^{ln x}=x for x>0. Use these definitions to evaluate logarithms, solve equations such as e^{2x}=7 by taking logs, and sketch or interpret graphs of ln x and e^x, noting their symmetry about y=x.
Understand and use the laws of logarithms: log_a(x) + log_a(y) ≡ log_a(xy); log_a(x) − log_a(y) ≡ log_a(x/y); k log_a(x) ≡ log_a(x^k) (including, for example, k = −1 and k = −1/2).
This item requires you to understand and apply the three logarithm laws for any positive base a, with x>0 and y>0. The addition law combines logs: log_a(x)+log_a(y)=log_a(xy). The subtraction law gives a quotient: log_a(x)−log_a(y)=log_a(x/y). The power law moves a coefficient: k log_a(x)=log_a(x^k), including negative and fractional k, so −log_a(x)=log_a(x^{−1})=log_a(1/x) and −½ log_a(x)=log_a(x^{−1/2})=log_a(1/√x). You use these to simplify expressions, expand or condense logarithmic forms, and solve equations such as log_2(x)+log_2(x−2)=3 by combining to log_2(x(x−2))=3, then converting to x(x−2)=2^3 and checking that solutions satisfy x>0 and x−2>0.
Solve equations of the form a^x = b.
An equation a^x=b has the unknown in the exponent, so ordinary rearrangement cannot isolate x. The standard method takes logarithms of both sides: log(a^x)=log b, then the power law gives x log a=log b, so x=log b/log a. Any consistent base works; base 10 or e is convenient on a calculator. For example, 3^x=20 gives x=log 20/log 3≈2.73. Check by substitution. If b is a power of a, or both sides can be written with the same base, comparing indices is quicker: 2^x=8 gives x=3 because 8=2^3. Recognise when b≤0: for a>0 there is no real solution, since a^x is always positive. State answers to the accuracy required, and keep full calculator display until the final rounding to avoid premature rounding errors.
Use logarithmic graphs to estimate parameters in relationships of the form y = axⁿ and y = kbˣ, given data for x and y.
When data follow y = axⁿ or y = kbˣ, plotting on logarithmic axes turns the curve into a straight line whose gradient and intercept give the unknown parameters. For y = axⁿ, take logarithms: log y = log a + n log x, so plotting log y against log x gives gradient n and intercept log a, hence a = 10^intercept. For y = kbˣ, log y = log k + x log b, so plotting log y against x gives gradient log b and intercept log k, hence k = 10^intercept and b = 10^gradient. Use the same logarithm base throughout, usually base 10. Estimate the gradient from two well-separated points on the line of best fit, then convert back by raising the base to the gradient or intercept. Always interpret the parameters in context and comment on how well the line fits the plotted points.
Understand and use exponential growth and decay; use in modelling (examples may include the use of e in continuous compound interest, radioactive decay, drug concentration decay, exponential growth as a model for population growth); consideration of limitations and refinements of exponential models.
Exponential models describe quantities changing at a rate proportional to their current size. You use y = A e^{kt} or y = A b^t, where A is the initial value and k (or ln b) is the continuous rate constant. For growth, k > 0; for decay, k < 0. In continuous compound interest, A is the principal, k the nominal annual rate, and t time in years. Radioactive decay uses half-life to find k = ln(0.5)/half-life. Drug concentration often decays exponentially after absorption. Population growth may be modelled with k = ln(growth factor per year). Limitations include finite resources, so refinements may use logistic models or piecewise exponential models. You must interpret parameters in context, solve for unknowns using logarithms, and critique assumptions such as constant rate, unlimited growth, or no external factors.
Your focus
- Sketch the graph of y=a^x for a>0, showing the correct intercept and asymptote (noting a=1 is constant).
- Explain the difference between the graphs of y=a^x for a>1, 0<a<1 and a=1.
- Sketch and interpret the graph of y=e^x, including its key features.
Show all 21 objectives
- Differentiate functions of the form e^(kx) correctly.
- Explain why the derivative being proportional to the function makes the exponential model suitable for growth and decay.
- Apply the derivative of e^(kx) to solve problems involving rates of change.
- Evaluate logarithms using the definition log_a(x) as the inverse of a^x.
- Sketch and interpret the graph of ln x, including its domain and asymptote.
- Solve equations by applying ln x as the inverse function of e^x.
- Apply the addition, subtraction and power laws of logarithms to simplify expressions.
- Use the laws to solve logarithmic equations and check for extraneous solutions.
- Handle negative and fractional coefficients in the power law correctly.
- Apply logarithms to both sides of an exponential equation and use the power law to isolate the unknown exponent.
- Evaluate the resulting quotient accurately and present the solution to an appropriate degree of accuracy.
- Identify equations with no real solution by reasoning about the range of a^x for positive a.
- Transform y = axⁿ and y = kbˣ into linear form using logarithms and identify the quantities to plot.
- Estimate the gradient and intercept from a logarithmic graph and convert them into the parameters a, n, k and b.
- Evaluate how well a straight-line model fits given data and communicate the resulting relationship in context.
- Students can set up an exponential model from a described situation and interpret its parameters.
- Students can solve problems involving exponential growth and decay, including finding unknown times or rates.
- Students can evaluate the limitations of an exponential model and suggest appropriate refinements.
F: Exponentials and logarithms exam tips
Marking Points
- State that for a>0, a^x is defined for all real x and is always positive.
- Identify the y-intercept as (0,1) for any positive base a.
- Describe the horizontal asymptote y=0 and the increasing/decreasing behaviour for a>1 and 0<a<1, and state that a=1 gives the constant function y=1 with no asymptote.
- Sketch y=e^x showing correct shape, intercept (0,1) and asymptotic behaviour as x→-∞.
- Apply transformations such as y=e^(x+2) or y=3e^x to the graph of y=e^x, working out key point coordinates.
- Use the graph to solve equations or inequalities involving a^x or e^x.
- Differentiate e^(kx) correctly as k e^(kx), applying the chain rule where the exponent is a function of x.
- Explain that the derivative is proportional to the original function, with constant of proportionality k.
- Interpret the sign of k as indicating growth (k > 0) or decay (k < 0), and state its meaning and units in context.
- Apply the result to contextual problems, such as finding the rate of change from a given exponential model or the gradient at a specific point.
- Recognise that the exponential model is appropriate when the rate of change is proportional to the current value, and use the derivative to form the corresponding differential equation.
- Explain that Newton's law of cooling fits this model when the variable is defined as the temperature difference from the surroundings, since then dy/dt = -ky.
- Correctly interprets log_a(x) as the exponent to which a must be raised to obtain x, and uses this to evaluate expressions such as log_2(8)=3.
- Applies inverse relationships: log_a(a^x)=x for all real x, and a^{log_a(x)}=x for x>0.
- States and uses the domain of log_a(x) as x>0, recognising that logarithms of zero or negative numbers are undefined in real numbers; notes that the specification's 'x ≥ 0' is corrected to x>0.
- Sketches or describes the graph of ln x, including shape, vertical asymptote at x=0, and key point (1,0).
- Uses ln x as the inverse of e^x to solve equations, for example solving e^{3x}=10 by writing 3x=ln 10.
- Explains the reflection relationship between the graphs of y=e^x and y=ln x in the line y=x.
- Applies the addition law correctly: log_a(x)+log_a(y)=log_a(xy), for x>0 and y>0.
- Applies the subtraction law correctly: log_a(x)−log_a(y)=log_a(x/y), for x>0 and y>0.
- Uses the power law k log_a(x)=log_a(x^k), including negative and fractional values of k, such as −1 and −1/2.
- Combines multiple laws to simplify or expand logarithmic expressions, for example writing 2 log_a(x) − log_a(y) as log_a(x^2/y).
- Solves logarithmic equations by first using the laws to combine terms, then converting to exponential form and checking domain restrictions.
- Takes logarithms of both sides of the equation, or otherwise correctly applies the power law for logarithms to bring the exponent down.
- Rearranges to x=log b/log a, or the equivalent using natural logarithms, showing the division by log a.
- Evaluates the quotient accurately, retaining sufficient figures before rounding to the accuracy requested.
- Where the equation can be written with a common base (for example b is a power of a), correctly equates the indices and solves the resulting linear equation.
- Recognises and states that no real solution exists when b is zero or negative for a positive base a.
- Checks the solution by substitution or by considering the approximate size of the answer.
- Takes logarithms of the given relationship and rearranges into the form Y = mX + c, identifying which quantities to plot.
- Plots the transformed data correctly, using log y against log x for y = axⁿ and log y against x for y = kbˣ.
- Draws a line of best fit and estimates its gradient using two well-separated points on the line.
- Interprets the gradient and intercept correctly: for y = axⁿ, gradient is n and intercept is log a; for y = kbˣ, gradient is log b and intercept is log k.
- Converts the intercept back to the parameter by raising the logarithm base to that value, and states the final relationship with the estimated parameters.
- Comments on the suitability of the model by referring to how closely the plotted points lie to the straight line.
- Recognise when a situation is modelled by exponential growth or decay and select an appropriate form such as y = A e^{kt} or y = A b^t.
- Interpret the parameters A and k (or b) in the context of the problem, including initial value, continuous rate, and growth/decay factor.
- Use given data to determine the model, for example by using two data points to form simultaneous equations or by using a half-life to find k.
- Solve problems involving exponential models, including finding times for a quantity to reach a given value, using logarithms where necessary.
- Critique exponential models by identifying limitations such as unrealistic long-term behaviour, and suggest refinements such as a logistic model or a restricted domain.
Examiner Tips
- 💡When sketching, mark the intercept (0,1) and, for a≠1, the asymptote y=0 to gain clarity and method marks.
- 💡For transformations, work out the coordinates of key points such as the intercept and one other point before drawing the curve.
- 💡If asked to compare a^x and e^x, comment on the base value and the resulting steepness rather than just repeating the shape.
- 💡Use the graph to justify solutions to equations; a rough sketch can confirm the number of intersections.
- 💡When asked to show that a model is suitable, differentiate the given function and show that the derivative is a constant multiple of the function.
- 💡In application questions, state the meaning of k clearly in the context of the problem, including its sign and units.
- 💡Use the derivative to find the rate of change at a given time by substituting the time into k e^(kx).
- 💡If the model is N = N0 e^(kt), remember that N0 is the initial value and k is the continuous rate constant.
- 💡When solving exponential equations, isolate the exponential term first, then take natural logarithms of both sides and use ln(e^u)=u.
- 💡For graph questions, mark the asymptote and at least two key points, such as (1,0) and (e,1) for y=ln x, and show the reflection in y=x if both graphs are required.
- 💡Check that any logarithm you evaluate or solve has a positive argument; state this restriction explicitly when it affects the solution set.
- 💡When simplifying, decide whether to condense to a single logarithm or expand fully; read the question to see which form is required.
- 💡For equations, combine logarithms on one side first, then rewrite in exponential form; always verify solutions in the original equation to reject any that make an argument non-positive.
- 💡Handle negative and fractional coefficients by writing them as powers, for example −½ log_a(x)=log_a(x^{−1/2}), and simplify the resulting index if needed.
- 💡Show the logarithmic step explicitly before evaluating, so the method is visible even if the final value is mis-keyed.
- 💡Use the same base for both logarithms; mixing log and ln within one calculation invites error.
- 💡Store the unrounded quotient in the calculator and round only at the end, then give the answer to the accuracy stated in the question.
- 💡Label the axes clearly as log y and log x, or log y and x, so the examiner can see which transformation you have used.
- 💡Show the substitution of your two chosen points into the gradient formula before evaluating, and keep the unrounded gradient for the conversion.
- 💡Finish by writing the relationship with numerical values for the parameters and, where the context is given, interpret what they mean.
- 💡Always define your variables and state the model you are using before substituting values.
- 💡When solving for time, take natural logarithms of both sides and show your working clearly.
- 💡In modelling questions, comment on the validity of the model in context, referring to the assumptions and possible refinements.
Common Mistakes
- Thinking that a^x can be negative for some x: correct this by noting that a positive base raised to any real power is positive.
- Drawing the graph crossing the x-axis: correct by showing the x-axis as an asymptote that the curve approaches but never meets (for a≠1).
- Treating a=1 as having the same asymptote as other bases: correct by noting y=1^x=1 is the constant line y=1, with no asymptote.
- Forgetting that e is a specific constant approximately 2.718, not a variable: correct by treating e^x as a particular exponential function.
- Differentiating e^(kx) as e^(kx) without multiplying by k: correct by applying the chain rule to get k e^(kx).
- Thinking that the gradient is constant: correct by noting that the gradient changes with x and equals k times the function value.
- Misinterpreting k as the growth rate per unit time in discrete terms: correct by explaining that k is the continuous rate constant, and the actual rate depends on the current value.
- Claiming Newton's law of cooling is not an exponential model: correct by defining y as the temperature difference from the surroundings, so that dy/dt = -ky and y decays exponentially.
- Assuming any dependence of the rate of change on the amount present makes an exponential model suitable: correct by checking that the rate is proportional to the amount, i.e. dy/dt = ky.
- Thinking log_a(x) can accept x=0 or negative x. Correction: the domain is x>0; log_a(0) and log_a(negative) are undefined in real numbers.
- Confusing log_a(x) with a^x or treating them as the same function. Correction: they are inverses, so log_a(a^x)=x and a^{log_a(x)}=x, but the functions themselves are different.
- Believing ln x is the inverse of 10^x or that ln means log base 10. Correction: ln x is log base e, and its inverse is e^x; log base 10 is written log x or log_10 x.
- Writing log_a(x+y) as log_a(x)+log_a(y). Correction: the addition law applies to the sum of two logarithms, not the logarithm of a sum; log_a(x+y) cannot be split in this way.
- Applying the power law incorrectly as k log_a(x)= (log_a(x))^k or as log_a(kx). Correction: k log_a(x)=log_a(x^k).
- Forgetting to check that arguments remain positive after combining or solving. Correction: always state and enforce x>0, y>0, and for combined expressions such as log_a(xy), require xy>0, which may impose conditions on x and y.
- Writing x=log b−log a instead of dividing: the power law gives x log a=log b, so the correction is to divide both sides by log a.
- Taking logarithms of only one side, which breaks the equality; the correction is to apply the same logarithm to both sides.
- Attempting to solve a^x=−5 and giving a real answer; the correction is to state that a positive base raised to any real power is positive, so there is no real solution.
- Stating the common-base condition backwards (for example 'a is a power of b'): correct by noting it is b that must be a power of a, or both sides must share a base.
- Plotting y against x on logarithmic axes but reading the gradient as the parameter directly; the correction is to convert the gradient back, for example b = 10^gradient for y = kbˣ.
- Forgetting to convert the intercept from log a or log k back to a or k; the correction is to raise the base to the intercept value.
- Using points that are close together to find the gradient, which magnifies reading error; the correction is to choose two points far apart on the line of best fit.
- Confusing the continuous rate k with the discrete growth factor b. Correction: remember b = e^k, so if a quantity doubles each year, b = 2 and k = ln 2 ≈ 0.693.
- Forgetting to convert time units or using inconsistent units. Correction: ensure t and k use the same time unit, for example if k is per year, t must be in years.
- Assuming exponential growth continues indefinitely without considering limiting factors. Correction: state that real populations have carrying capacity, so a logistic model may be more appropriate for long-term predictions.