Exponentials and logarithms — Edexcel A-Level Mathematics
Test yourself on Exponentials and logarithms with PEARSON EDEXCEL A-Level practice questions.
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Exponentials and logarithms explained
An exponential function has the variable in the exponent: y = a^x with a > 0.
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Its graph always passes through (0, 1) because a⁰ = 1, stays entirely above the x-axis since a^x > 0 for all real x, and has the x-axis as a horizontal asymptote. If a > 1 the graph increases; if 0 < a < 1 it decreases, and the two cases are reflections of each other in the y-axis. The special base e ≈ 2.718 is defined so that the curve y = e^x has gradient equal to its own value at every point. To sketch, plot (0, 1), (1, a) and a second guide point such as (−1, 1/a), then draw a smooth curve approaching but never touching the axis. Recognise transformations: y = a^(x + c) shifts left by c, y = a^x + d shifts up by d, and y = a^(kx) stretches horizontally.
6.2 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): the function reproduces itself multiplied by the constant k. This follows from the chain rule, since the derivative of e^u with respect to u is e^u and u = kx has derivative k. The result means the rate of change is proportional to the current value, dy/dx = ky, which is exactly the relationship modelled by growth and decay processes such as populations, compound interest, radioactive decay and cooling. When k > 0 the quantity grows; when k < 0 it decays. You can verify the rule numerically by comparing gradients from a table of values, and you use it to find gradients, tangents and stationary behaviour of exponential curves.
6.3 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.
The logarithm log_a(x) answers the question: to what power must a be raised to give x? So y = log_a(x) exactly when a^y = x, making log_a the inverse of a^x. Because a^x is always positive, the domain of log_a(x) is x > 0, and its graph is the reflection of y = a^x in the line y = x, passing through (1, 0) with the y-axis as a vertical asymptote. The natural logarithm ln x is log_e(x), the inverse of e^x, so ln(e^x) = x and e^(ln x) = x for x > 0. Its graph passes through (1, 0) and (e, 1) and rises ever more slowly. These inverse relationships let you solve equations by taking logs or exponentiating both sides.
6.4 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).
The logarithm laws follow from the index laws because logs are exponents. Adding logs of the same base multiplies the arguments: log_a(x) + log_a(y) = log_a(xy). Subtracting logs divides the arguments: log_a(x) − log_a(y) = log_a(x/y). A coefficient becomes a power: k log_a(x) = log_a(x^k), which works for any real k, including k = −1 giving −log_a(x) = log_a(x^(−1)) = log_a(1/x), and k = −1/2 giving −(1/2)log_a(x) = log_a(x^(−1/2)) = log_a(1/√x). These laws let you combine or split logarithms, solve logarithmic equations by condensing to a single log, and change the subject in exponential models. They apply only when all logs share the same base and all arguments are positive.
6.5 Solve equations of the form a^x = b.
An equation aˣ = b has the unknown in the exponent, so ordinary rearrangement cannot isolate x. Take logarithms of both sides: log(aˣ) = log b, then use the power law to bring x down, giving x log a = log b and hence x = log b ÷ log a. Any consistent base works; choosing base 10 or base e suits most calculators. For example, 3ˣ = 20 gives x = log 20 ÷ log 3 ≈ 1.3010 ÷ 0.4771 ≈ 2.73. Check by substitution: 3^2.73 ≈ 20. If b is a recognisable power of a, equate indices directly, as with 2ˣ = 32 giving x = 5. Equations such as 5^(2x−1) = 40 are solved the same way, keeping the whole exponent intact before dividing.
6.6 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, taking logarithms gives log y = log a + n log x, so plotting log y against log x gives a straight line with gradient n and intercept log a. When data follow y = kbˣ, taking logarithms gives log y = log k + x log b, so plotting log y against x gives gradient log b and intercept log k. Estimate the gradient from two well-separated points on the line of best fit, then find the intercept where the line crosses the vertical axis. Convert back by exponentiating: a = 10^(intercept) for base 10, or a = e^(intercept) for natural logs. For example, if log y = 1.2 + 0.8 log x, then n = 0.8 and a = 10^1.2 ≈ 15.8, so y ≈ 15.8x^0.8.
6.7 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 growth and decay describe quantities changing at a rate proportional to their current size, modelled by y = Ae^(kt) or y = Ab^t. Growth occurs when k is positive, decay when k is negative. Continuous compound interest uses A = Pe^(rt), where P is the principal, r the annual rate and t the time in years. Radioactive decay uses N = N₀e^(−λt), with λ the decay constant and half-life t½ = ln 2 ÷ λ. Drug concentration often falls as C = C₀e^(−kt), and population growth may follow P = P₀e^(rt). Models assume constant rates and unlimited resources, so they have limitations: real populations face resource limits, drug clearance varies between patients, and decay rates may change. Refinements include logistic growth or piecewise models.
Your focus
- Sketch the graph of y = a^x for a given positive base, showing the intercept and asymptote.
- Explain why a^x is positive for all real x and how the base determines increasing or decreasing behaviour.
- Sketch and interpret the graph of y = e^x, including the significance of the base e.
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- Differentiate functions of the form e^(kx) accurately, including cases with negative k.
- Explain how dy/dx = ky arises from the derivative of e^(kx) and what it means physically.
- Apply the derivative to find gradients, tangents and to interpret growth or decay contexts.
- Convert fluently between a^y = x and y = log_a(x) and use this to evaluate logarithms.
- Sketch y = ln x and y = log_a(x), showing intercept, asymptote and relationship to the exponential graph.
- Apply the inverse identities involving ln x and e^x to solve equations.
- Apply the product, quotient and power laws to combine and split logarithms of the same base.
- Handle negative and fractional coefficients by converting them into powers of the argument.
- Solve logarithmic equations using the laws and verify that all solutions satisfy the domain.
- Take logarithms of both sides of an equation of the form aˣ = b.
- Apply the power law to isolate the unknown exponent.
- Evaluate and verify the solution in the original equation.
- Convert power and exponential relationships into linear logarithmic form.
- Estimate gradient and intercept from a logarithmic graph.
- Recover the original parameters and state the fitted relationship.
- Construct exponential growth and decay models from given contexts.
- Evaluate models at specified times and interpret the parameters.
- Critically discuss limitations and suggest refinements of exponential models.
Exponentials and logarithms exam tips
Marking Points
- States that a^x is defined for all real x provided a > 0, and that a^x is always positive, so the graph lies wholly above the x-axis.
- Identifies the y-intercept (0, 1) for every valid base and explains it follows from a⁰ = 1.
- Distinguishes increasing behaviour when a > 1 from decreasing behaviour when 0 < a < 1, linking this to the size of the base.
- Describes the x-axis as a horizontal asymptote and explains that the curve approaches it as x → −∞ (for a > 1) without meeting it.
- Sketches y = e^x with correct shape, intercept and asymptote, and states e ≈ 2.718 as the base making the gradient equal to the function value.
- Applies transformations correctly, for example reading y = e^(x − 2) + 3 as a translation of y = e^x by 2 right and 3 up.
- States the derivative of e^(kx) as k e^(kx) and identifies k as a constant multiplier arising from the chain rule.
- Shows the chain-rule reasoning: differentiate the outer exponential, then multiply by the derivative of the inner function kx.
- Links the result to the differential equation dy/dx = ky, explaining that rate of change is proportional to the quantity present.
- Interprets the sign of k: positive k gives growth, negative k gives decay, and connects this to real contexts such as population or radioactive decay.
- Applies the rule to find gradients, equations of tangents or normals, and to locate and classify stationary points where they exist.
- Uses the rule with a composite inner function, for example differentiating e^(3x + 1) by recognising k = 3.
- Defines log_a(x) as the power to which a must be raised to obtain x, and rewrites between forms a^y = x and y = log_a(x).
- Explains that log_a is the inverse function of a^x, so the two graphs are reflections in y = x.
- States the domain x > 0 and identifies the vertical asymptote at x = 0, with intercept (1, 0).
- Defines ln x as log_e(x) and uses the inverse identities ln(e^x) = x and e^(ln x) = x for x > 0.
- Sketches y = ln x with correct shape, intercept and asymptote, and relates it to the graph of y = e^x.
- Uses the inverse relationship to solve equations, for example converting e^(2x) = 7 into 2x = ln 7.
- States the product law log_a(x) + log_a(y) = log_a(xy) and applies it to combine two logarithms of the same base.
- States the quotient law log_a(x) − log_a(y) = log_a(x/y) and applies it to simplify differences of logarithms.
- States the power law k log_a(x) = log_a(x^k) and uses it to move coefficients into the exponent.
- Handles negative and fractional coefficients, showing −log_a(x) = log_a(1/x) and −(1/2)log_a(x) = log_a(1/√x).
- Uses the laws to solve equations, for example condensing log_a(x) + log_a(x − 3) into log_a(x(x − 3)) before removing the logarithm.
- Checks that all arguments remain positive and that every logarithm in a calculation has the same base.
- Takes logarithms of both sides of aˣ = b using a consistent base, for example log₁₀ or ln.
- Applies the power law log(aˣ) = x log a to bring the exponent down as a product.
- Rearranges to x = log b ÷ log a and evaluates to a sensible number of significant figures.
- Recognises cases where b is an exact power of a and equates indices instead, for example 2ˣ = 32 giving x = 5.
- Handles a compound exponent such as 5^(2x−1) = 40 by keeping the exponent intact, then solving the resulting linear equation.
- Checks the solution by substituting back into the original equation and comparing values.
- Takes logarithms of both sides to convert y = axⁿ into log y = log a + n log x.
- Takes logarithms of both sides to convert y = kbˣ into log y = log k + x log b.
- Plots the correct transformed variables, log y against log x or log y against x, and draws a line of best fit.
- Estimates the gradient from two well-separated points on the line and interprets it as n or log b.
- Reads the vertical intercept and converts it back to a or k by exponentiating with the chosen base.
- States the final relationship with the estimated parameters and comments on how well the line fits the data.
- Sets up an exponential model of the form y = Ae^(kt) or y = Ab^t with parameters identified from the context.
- Interprets the sign of the rate constant: positive for growth, negative for decay.
- Uses the continuous compound interest formula A = Pe^(rt) correctly, keeping t in years and r as a decimal.
- Applies radioactive decay N = N₀e^(−λt) and links the decay constant to half-life using t½ = ln 2 ÷ λ.
- Models drug concentration decay or population growth with an appropriate exponential function and evaluates at a given time.
- Discusses limitations such as constant rate assumptions, resource limits or individual variation, and suggests a refinement such as a logistic model.
Examiner Tips
- 💡Always mark the intercept (0, 1) and the asymptote on any exponential sketch, as these features carry the meaning of the curve.
- 💡When a question gives a point on y = a^x, substitute the coordinates and solve for a, checking that your value is positive.
- 💡For transformed exponentials, identify the base curve first, then apply each translation or stretch in turn to avoid combining them incorrectly.
- 💡Use a quick table of three or four values to confirm the shape before drawing, especially when the base is a fraction.
- 💡Write down k explicitly before differentiating, especially when the exponent is a compound expression such as 4x − 7.
- 💡If a question describes a rate proportional to the amount present, translate it immediately into dy/dx = ky and identify k from the context.
- 💡Check the sign of your derivative against the expected growth or decay described in the question.
- 💡When finding a tangent, evaluate both the function and its derivative at the given x-value before forming the line equation.
- 💡Convert between exponential and logarithmic form before attempting to solve, as this often reveals the required operation.
- 💡Mark the intercept (1, 0) and the vertical asymptote whenever you sketch y = ln x or y = log_a(x).
- 💡Use ln to undo e and e to undo ln, applying the same operation to both sides of an equation.
- 💡Check that any solution lies in the domain x > 0 before stating it as final.
- 💡Write the base alongside every logarithm so you notice immediately if bases differ.
- 💡Condense multiple logarithms into one before exponentiating both sides to solve an equation.
- 💡When a coefficient is negative or fractional, convert it to a power of the argument rather than moving it incorrectly.
- 💡Substitute your final answer back into the original equation to confirm every argument is positive.
- 💡State the logarithm step explicitly before using the power law, so the method is visible.
- 💡Give the final answer to three significant figures unless the question specifies otherwise.
- 💡Substitute the answer back into the original equation as a quick numerical check.
- 💡Label the transformed axes clearly, including the logarithm base used.
- 💡Choose two points far apart on the line of best fit to reduce gradient error.
- 💡Show the conversion from intercept to parameter explicitly, including the base.
- 💡Define each symbol and its units before substituting values into a model.
- 💡Keep full calculator accuracy through intermediate steps and round only the final answer.
- 💡When asked about limitations, link each limitation to a specific assumption of the model.
Common Mistakes
- Treating a^x as a polynomial and drawing a parabola or straight line; correct this by tabulating values such as a⁰, a¹ and a^(−1) and plotting a smooth curve with an asymptote.
- Believing the graph can cross or touch the x-axis for some x; correct this by noting a^x > 0 for every real x when a > 0.
- Assuming every exponential graph increases; correct this by checking whether the base lies above or below 1, since 0 < a < 1 gives a decreasing curve.
- Confusing the roles of base and exponent, for example reading 2^x as x²; correct this by stressing that in a^x the variable is the exponent.
- Forgetting to multiply by k and writing the derivative as e^(kx); correct this by applying the chain rule fully so the factor k appears.
- Differentiating the exponent as though it were a power of x, for example treating e^(kx) like x^n; correct this by keeping the exponential form unchanged.
- Losing the negative sign when k is negative, which reverses growth and decay; correct this by substituting k carefully, for example k = −0.05 gives −0.05e^(−0.05x).
- Assuming every exponential curve has a stationary point; correct this by noting k e^(kx) is never zero for finite x, so no turning points occur.
- Writing log_a(x) for negative or zero x; correct this by stating the domain x > 0 and rejecting invalid inputs.
- Treating ln x as a power function such as x^n; correct this by sketching the logarithmic curve with its asymptote and slow growth.
- Confusing log_a(x) with (log a) × x or with a^x; correct this by translating to the exponent form a^y = x.
- Applying ln to a sum term by term, for example replacing ln(x + 3) with ln x + ln 3; correct this by noting no such law exists for sums.
- Applying the product law to a sum inside one logarithm, for example rewriting log_a(x + y) as log_a(x) + log_a(y); correct this by keeping log_a(x + y) unchanged.
- Using the laws across different bases, such as combining log_2(x) with log_3(y); correct this by converting to a common base first.
- Dropping the negative sign when k is negative, for example writing −log_a(x) as log_a(x); correct this by writing log_a(1/x).
- Forgetting to reject solutions that make an argument zero or negative after solving; correct this by substituting back into the original equation.
- Dividing b by a instead of taking logarithms; correct this by taking logs of both sides first.
- Writing log b ÷ log a as log(b ÷ a); correct this by keeping the two logarithms separate.
- Dropping the exponent when the base is a power of the other side; correct this by equating indices only when the bases match exactly.
- Plotting y against x instead of the transformed variables; correct this by plotting log y against log x or log y against x.
- Confusing the gradient with the intercept; correct this by identifying which parameter multiplies the variable and which is the constant term.
- Forgetting to exponentiate the intercept; correct this by converting log a or log k back to a or k.
- Using a percentage rate directly as the exponent without converting to a decimal; correct this by writing r as a decimal, for example 5% as 0.05.
- Treating a decay constant as positive in the exponent; correct this by using a negative exponent for decay.
- Assuming exponential growth continues indefinitely; correct this by discussing resource limits and refinements such as logistic growth.