Question

solve for x in the equation below. round your answer to the nearest hundredth. do not round any intermediate computations. $9^{x - 5} = 4$

Explanation:

Step1: Take log of both sides

Take the logarithm (natural log or common log) of both sides. Let's use natural log ($\ln$). So, $\ln(9^{x - 5})=\ln(4)$.

Step2: Apply log power rule

Using the power rule of logarithms, $\ln(a^b)=b\ln(a)$, we get $(x - 5)\ln(9)=\ln(4)$.

Step3: Solve for x

First, divide both sides by $\ln(9)$: $x - 5=\frac{\ln(4)}{\ln(9)}$. Then, add 5 to both sides: $x = 5+\frac{\ln(4)}{\ln(9)}$. Calculate the value: $\frac{\ln(4)}{\ln(9)}\approx\frac{1.3863}{2.1972}\approx0.6315$. So, $x\approx5 + 0.6315 = 5.6315\approx5.63$.

Answer:

$x\approx5.63$