How to Convert an Exponential Expression to a Different Base

To rewrite an expression with one exponential base using another base, use logarithms. The general identity is

$$a^x=b^{x\log_b a}.$$

This works for positive $a$ and $b$, where $b\ne 1$. It is often quicker to use ordinary powers when the original base is an exact power of the new base.

The base-conversion rule

Suppose you want to express $a^x$ as a power of $b$. Let the unknown exponent be $y$, so that $a^x=b^y$. Taking logarithms to base $b$ gives

$$y=\log_b(a^x)=x\log_b a.$$

Therefore,

$$a^x=b^{x\log_b a}.$$

The logarithm factor changes the exponent to compensate for changing the base. You do not generally replace the base and leave the exponent unchanged.

The change-of-base formula lets you calculate the factor using another logarithm, such as a natural logarithm or a common logarithm:

$$\log_b a=\frac{\ln a}{\ln b}=\frac{\log_{10}a}{\log_{10}b}.$$

Thus an equivalent version of the conversion rule is

$$a^x=b^{x\frac{\ln a}{\ln b}}.$$

Worked examples

Converting 8^x to base 2

Because $8=2^3$, substitute that power into the expression and use the exponent rule $(u^m)^n=u^{mn}$:

$$8^x=(2^3)^x=2^{3x}.$$

The exponent is multiplied by 3 because 8 is the third power of 2.

Converting 5^3 to base 2

Since 5 is not an integer power of 2, use a logarithm:

$$5^3=2^{3\log_2 5}.$$

For a numerical check, $\log_2 5\approx2.321928$, so the exponent is approximately $6.965784$. Then $2^{6.965784}$ is approximately 125, which agrees with $5^3=125$.

Converting e^(kx) to base 10

Set $a=e$, $b=10$, and use $x$ in the general rule as $kx$:

$$e^{kx}=10^{kx\log_{10}e}=10^{kx/\ln 10}.$$

This form is useful when you need an exponential expression written using powers of 10, for example to compare it with another base-10 exponential.

A shortcut when the bases are related

If the old base is an exact power of the new base, first rewrite the old base and then simplify. For example,

$$16^{x+1}=(2^4)^{x+1}=2^{4x+4}.$$

You can also convert in the other direction. Since $2=16^{1/4}$,

$$2^x=16^{x/4}.$$

The logarithm rule gives the same result: $\log_{16}2=1/4$.

Common mistakes

• Changing the base but not the exponent: In general, $a^x$ is not equal to $b^x$. The exponent needs the factor $\log_b a$.
• Using the logarithm in the wrong order: To convert $a^x$ into base $b$, use $\log_b a$, not $\log_a b$. These are reciprocals when both are defined.
• Forgetting the whole exponent: If the original expression is $a^{f(x)}$, the converted exponent is $f(x)\log_b a$. For example, $8^{x+1}=2^{3(x+1)}$, not $2^{3x+1}$.
• Ignoring domain conditions: The real logarithm $\log_b a$ requires $a>0$, $b>0$, and $b\ne1$. Expressions with negative bases and non-integer exponents need separate care and do not follow this real-logarithm rule in general.

Converting a numerical exponential to a calculator-friendly base

If you need a numerical value, you usually do not have to rewrite the expression. A calculator can evaluate $a^x$ directly, or you can use $a^x=e^{x\ln a}$. The latter follows from the same identity and is useful in algebra, calculus, and programming when natural exponentials are the available function.