F: Exponentials and logarithms: 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 idea

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.

What earns the marks

• 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.

Common mistakes and how to put them right

• Thinking log_a(x) can accept x=0 or negative x.

  Put it right: 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.

  Put it right: 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.

  Put it right: Ln x is log base e, and its inverse is e^x; log base 10 is written log x or log_10 x.

Exam tips

• 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.