Question

solve. round your answer to the nearest thousandth.
$22 = 5^{3x}$
$x = \square$

Explanation:

Step1: Take log of both sides

Take the common logarithm (base 10) of both sides of the equation \(22 = 5^{3x}\). We get \(\log(22)=\log(5^{3x})\).

Step2: Apply logarithm power rule

Using the power rule of logarithms \(\log(a^b)=b\log(a)\), the right - hand side becomes \(3x\log(5)\). So the equation is \(\log(22) = 3x\log(5)\).

Step3: Solve for x

First, we can express \(x\) as \(x=\frac{\log(22)}{3\log(5)}\).
We know that \(\log(22)\approx1.3424\) and \(\log(5)\approx0.6990\).
Substitute these values into the formula for \(x\):
\(3\log(5)=3\times0.6990 = 2.097\)
\(x=\frac{1.3424}{2.097}\approx0.640\)

Answer:

\(x\approx0.640\)