Exponentials and logarithms

Learn what an exponential function is, how the base $a$ determines whether the function grows or decays, and what the graph of $f(x) = a^x$ looks like.

Understand $e \approx 2.718$, the limiting value of $(1 + 1/n)^n$, and the natural exponential function $e^x$ that underpins growth models across mathematics and machine learning.

Learn what $\log_a(x)$ means as the inverse of $a^x$, how to convert between exponential and logarithmic form, and how to evaluate logarithms like $\log_3(27)$ by inspection.

Learn how to simplify expressions using the two inverse identities $\log_a(a^x) = x$ and $a^{\log_a(x)} = x$, recognising when a composition of a logarithm and its matching exponential cancels.

Learn the common logarithm $\log(x) = \log_{10}(x)$, the natural logarithm $\ln(x) = \log_e(x)$, and how to use the inverse identities $\ln(e^x) = x$ and $e^{\ln x} = x$ to simplify expressions.

Learn how to convert products and quotients inside a logarithm into sums and differences of logarithms using the product and quotient rules, and bring exponents down as coefficients using the power rule.

Learn how to apply the product, quotient, and power rules to expand a single logarithm into a sum or difference, combine several logarithms into one, and simplify expressions that mix logs with $e^x$ and $\ln$.

Learn how to convert a logarithm from one base to another using the change of base formula, use the reciprocal relationship to relate logs with swapped bases, and relate logs with different bases when one is a power or root of the other.

Learn how to solve exponential equations for an unknown exponent by applying the matching logarithm when the base allows it, or by taking the logarithm of both sides and using the power rule to pull the exponent down.