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Equations with x in the exponent

When the unknown is in the exponent, take the logarithm of both sides and use $\lg b^x = x\lg b$ . First isolate the power so that it stands alone. If both sides are powers with the same base, you can instead set the exponents equal. A logarithmic equation can only have solutions that make the arguments positive.

Try it yourself

Exponential growth

a\cdot b^x = c \Rightarrow x = \frac{\lg(c/a)}{\lg b}

the solution

b^x = b^k \Rightarrow x = k

the same base

Symbols

$b$	the base, positive and not 1
$x$	the exponent

Example

$500\cdot 1.04^x = 800$ gives $1.04^x = 1.6$ and $x = \dfrac{\lg 1.6}{\lg 1.04} \approx 12.0$ . $2^x = 32$ gives $x = 5$ , since $2^5 = 32$ .

Divide out the constant in front of the power before you take the logarithm.

Practise powers, roots and logarithms for free →

← Common logarithms · Growth factor →

Part of Mathematics 1T: Powers, roots and logarithms.