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A Levels / Maths

Exponentials and Logarithms

Learn A-Level exponentials and logarithms, including exponential graphs, inverse functions, natural logarithms, log laws, exponential equations, logarithmic graphs, growth and decay, and mathematical modelling.

An exponential function has the form y=axy=a^x, where a>0a>0, and its exponential graph changes according to the value of the base aa.
When graphing an exponential function such as y=axy=a^x, the graph passes through (0,1)(0,1) because a0=1a^0=1, and the xx-axis acts as a horizontal asymptote.
The function y=exy=e^x is an important exponential function, where ee is the natural exponential base, and its gradient at every point is equal to its function value.
An exponential graph for y=exy=e^x is always positive and increasing, and the fact that its derivative is also exe^x makes it especially important in calculus and mathematical modelling.
Plotting exponential functions helps show rapid growth when the base is greater than 1 and decay when the base lies between 0 and 1.
A logarithm is the inverse operation of exponentiation: log⁡ax=y\log_a x=y means exactly that ay=xa^y=x, where a>0a>0, a≠1a\ne1 and x>0x>0.
Logarithmic functions are inverse functions of exponential functions, so the graphs of logarithms are reflections of their corresponding exponential graphs in the line y=xy=x.
The natural logarithm ln⁡x\ln x is the inverse function of exe^x, giving ln⁡(ex)=x\ln(e^x)=x and eln⁡x=xe^{\ln x}=x for x>0x>0.
A log graph such as y=ln⁡xy=\ln x is defined only for x>0x>0, passes through (1,0)(1,0) and has a vertical asymptote at x=0x=0.
The log laws include the product law log⁡a(xy)=log⁡ax+log⁡ay\log_a(xy)=\log_a x+\log_a y, which converts multiplication inside a logarithm into addition.
Another of the laws of logs is the quotient law log⁡a(x/y)=log⁡ax−log⁡ay\log_a(x/y)=\log_a x-\log_a y, which converts division inside a logarithm into subtraction.
The power law of logarithms is log⁡a(xk)=klog⁡ax\log_a(x^k)=k\log_a x, and these log rules allow logarithmic expressions to be expanded, simplified and combined.
Exponential equations of the form ax=ba^x=b can be solved using logarithms, giving x=log⁡blog⁡ax=\frac{\log b}{\log a} or equivalently x=ln⁡bln⁡ax=\frac{\ln b}{\ln a}.
Graphs of logarithms can be used to transform relationships such as y=axny=ax^n and y=kbxy=kb^x into straight-line forms, allowing unknown parameters to be estimated from data.
Exponential relationships are used to model growth and decay, including compound interest, population growth, radioactive decay and drug concentration, although the assumptions and limitations of each exponential model must be considered.
Exponentials and Logarithms | Exponential Graphs & Log Laws