How Do You Change the Base of a Logarithm?

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Most often, choose base $10$ or base $e$ because a GDC provides $\log$ and $\ln$ keys. This lets you evaluate logarithms whose bases are not available directly.

Why the formula works

Let $y=\log_b x$. By the definition of a logarithm, this means $b^y=x$. Take logarithms to base $c$ on both sides:

Apply the power law $\log_c(a^k)=k\log_c a$:

Dividing by $\log_c b$ gives the formula. The numerator contains the original argument, while the denominator contains the original base. The conditions are $x>0$, $b>0$, $c>0$, $b\neq1$, and $c\neq1$, ensuring every logarithm is defined and the denominator is non-zero.

Worked examples

To evaluate $\log_2 7$ using natural logarithms, write

This is reasonable because $2^2<7<2^3$, so the answer must lie between $2$ and $3$. This estimate helps detect an inverted fraction or calculator-entry error.

Change of base can also give an exact result:

It also solves exponential equations. If $5^x=12$, take logarithms and isolate the exponent:

Common misconception

A common misconception is reversing the fraction. The correct form places the logarithm of the argument over the logarithm of the base. Another mistake is splitting the logarithm of a sum; changing base does not justify false logarithm laws.

IB exam technique

This topic is IB Mathematics AA SL 1.7: Laws of Exponents and Logarithms and is also required at HL. On Paper 1, show the conversion and retain an exact form when appropriate. On Paper 2, write the converted expression before using your GDC, then give the answer to $3$ significant figures unless instructed otherwise.

For the command term Calculate, include the converted expression and relevant stages, not only the decimal display. In exact questions, do not replace a rational answer with an unnecessary approximation. Always check domain conditions, especially when variables appear in the argument or base. This protects method marks.