Question

solve for x in the equation below. round your answer to the nearest hundredth. do not round any intermediate computations. $6^{9x}=8$

Explanation:

Step1: Take natural log of both sides

$\ln(6^{9x}) = \ln(8)$

Step2: Use logarithm power rule

$9x\ln(6) = \ln(8)$

Step3: Solve for x

$x = \frac{\ln(8)}{9\ln(6)}$

Step4: Calculate the value

First, find $\ln(8)\approx2.079441542$, $\ln(6)\approx1.791759469$. Then $9\ln(6)\approx16.12583522$. So $x\approx\frac{2.079441542}{16.12583522}\approx0.129$ (rounded to nearest hundredth is 0.13). Wait, let's recalculate more accurately. $\ln(8)=2\ln(2)\approx2\times0.69314718056 = 1.38629436112$? Wait no, $\ln(8)=\ln(2^3)=3\ln(2)\approx3\times0.69314718056 = 2.07944154168$. $\ln(6)=\ln(2\times3)=\ln(2)+\ln(3)\approx0.69314718056 + 1.09861228866 = 1.79175946922$. Then $9\ln(6)=9\times1.79175946922 = 16.12583522298$. Then $\frac{\ln(8)}{9\ln(6)}=\frac{2.07944154168}{16.12583522298}\approx0.129$. Wait, 0.129 to nearest hundredth: look at thousandth place, 9, so round up the hundredth place: 0.13? Wait 0.129 is closer to 0.13? Wait 0.129: the hundredth digit is 2, thousandth is 9, so 0.129 ≈ 0.13 when rounded to nearest hundredth. Wait, let's check with calculator steps again. Alternatively, use common logarithm: $\log(6^{9x})=\log(8)\implies9x\log(6)=\log(8)\implies x=\frac{\log(8)}{9\log(6)}$. $\log(8)\approx0.903089987$, $\log(6)\approx0.7781512504$, $9\log(6)\approx7.003361253$, so $x\approx\frac{0.903089987}{7.003361253}\approx0.129$. Same result. So 0.129 rounded to nearest hundredth is 0.13? Wait 0.129: the number is 0.12 (hundredth place) and 9 in thousandth. So when rounding to nearest hundredth, since thousandth is 9 (≥5), we add 1 to hundredth place: 0.13.

Wait, maybe I made a mistake in ln(8). Wait no, $\ln(8)$ is indeed 3ln(2)≈2.07944. So the calculation is correct. So x≈0.13 when rounded to nearest hundredth.

Answer:

0.13