How Do You Evaluate a Logarithm With Base 10 or Base e Using Technology?

Use the log key to evaluate a base-$10$ logarithm and the ln key to evaluate a base-$e$ logarithm. Enter the positive argument, close the bracket if necessary, and press ENTER or =.

This is part of IB Mathematics: Analysis and Approaches (AA), SL 1.5: Introduction to logarithms. Numerical evaluation using technology is especially relevant to Paper 2.

Which Key Should You Use?

A logarithm represents an exponent. The expression $\log_{10}x$ asks which power of $10$ produces $x$. The natural logarithm $\ln x$ asks the same question using base $e$.

Worked Examples

For an exact value,

$$\log(1000)=3$$

because $10^3=1000$. Similarly,

$$\ln(e^2)=2$$

because logarithms reverse exponentiation.

For values without a simple exact form, enter the complete expression before rounding. A GDC gives

$$\log(7)=0.845098\ldots \approx 0.845$$

and

$$\ln(7)=1.94591\ldots \approx 1.95.$$

Both final answers are given to three significant figures, following the usual IB accuracy convention unless another degree of accuracy is specified. Keep the unrounded GDC value for any later calculation.

The argument of a real logarithm must satisfy $x>0$. Therefore, entering $\log(0)$, $\ln(0)$, or a logarithm of a negative number produces a domain error in real-number mode.

A common misconception is that ln means “log multiplied by $n$.” This is incorrect: $\ln x$ specifically means the logarithm of $x$ to base $e$. You can check a result by raising the relevant base to the displayed answer.

If the GDC has no key for another base, use the change-of-base formula:

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

Exam Technique

On Paper 2, write the logarithmic expression before recording the GDC result, and round only the final answer. On Paper 1, calculators are prohibited, so use exact logarithmic relationships and exponent laws instead.