What's the Difference Between Log and Ln?

In IB Mathematics: Analysis and Approaches, $\log x$ normally means a logarithm with base $10$, while $\ln x$ means a logarithm with base $e$, where $e \approx 2.718$. Both answer the same question: “To what power must the base be raised to obtain this number?”

A logarithm is the inverse of exponentiation. In general,

$$\log_a b=c \quad \Longleftrightarrow \quad a^c=b.$$

The notation changes according to the base:

The choice should match the exponential base. For $10^x=7$, applying $\log$ gives $x=\log 7$; for $e^x=7$, applying gives . Either type can solve either equation using the :

$$\log_a x=\frac{\ln x}{\ln a}.$$

For example, $\log_2 8=\frac{\ln 8}{\ln 2}=3$. A valid logarithm requires , , and . Graphically, and are inverse functions, so their graphs are reflections in . The same relationship connects and .

Although using the matching logarithm is usually simplest, the numerical answer does not depend on that choice. For example, taking $\log$ instead of $\ln$ in $e^{2x}=7$ gives $2x\log e=\log 7$, so . This equals by change of base. On a calculator, use the dedicated key for base and the key for base . For another base, enter a change-of-base quotient unless the GDC provides a direct logarithm template. Keep brackets around the entire numerator and denominator.

For example, solve $e^{2x}=7$. Applying $\ln$ to both sides gives

$$2x=\ln 7 \quad \Rightarrow \quad x=\frac{\ln 7}{2}.$$

A common misconception is that $\ln$ is a different operation from $\log$. It is not; it is simply a logarithm with base $e$. Also, $\log(x+y)\neq\log x+\log y$; logarithm laws apply to products, quotients, and powers, not sums.

Exam technique: This is AA SL 1.5, Intro to logs. On Paper 1, leave exact answers such as $\frac{\ln 7}{2}$. On Paper 2, use the GDC only when a numerical value is required. Always match the logarithm to the exponential base and check that every argument is positive.