To find the value of [tex]\( x \)[/tex] given the equation [tex]\( \log 9^x = 1.5 \)[/tex], follow these steps:
1. Simplify the logarithmic expression:
We know from logarithm properties that:
[tex]\[
\log a^b = b \log a
\][/tex]
Therefore, we can rewrite the given equation as:
[tex]\[
\log 9^x = x \log 9
\][/tex]
Given that [tex]\(\log 9^x = 1.5\)[/tex], we substitute to get:
[tex]\[
x \log 9 = 1.5
\][/tex]
2. Solve for [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], divide both sides by [tex]\( \log 9 \)[/tex]:
[tex]\[
x = \frac{1.5}{\log 9}
\][/tex]
3. Evaluate [tex]\( \log 9 \)[/tex]:
We know that [tex]\( 9 \)[/tex] can be expressed as [tex]\( 9 = 3^2 \)[/tex]. Using the logarithm properties again:
[tex]\[
\log 9 = \log (3^2) = 2 \log 3
\][/tex]
4. Find the numerical value:
The base-10 logarithm of 3 is approximately 0.4771. Therefore:
[tex]\[
\log 9 = 2 \log 3 \approx 2 \times 0.4771 = 0.9542
\][/tex]
5. Calculate [tex]\( x \)[/tex]:
Now substitute [tex]\( \log 9 \approx 0.9542 \)[/tex] back into the equation for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{1.5}{0.9542} \approx 1.5719274557170384
\][/tex]
6. Compare with the given options:
The value [tex]\( 1.5719274557170384 \)[/tex] needs to be matched with one of the provided options. The available choices are:
(a) 72
(b) 27
(c) 36
(d) 3.5
(e) 24.5
Comparing [tex]\( 1.5719274557170384 \)[/tex] with the given options, we observe that it does not exactly match any of them. Hence, none of the provided options are close to the computed value for [tex]\( x \)[/tex].
Since none of the options match our computed value closely, there might be a mistake in the options themselves. However, given the calculations, [tex]\( x \approx 1.5719274557170384 \)[/tex] is the solution.
Given that there must be a match:
None of the provided options (72, 27, 36, 3.5, 24.5) are correct based on the calculations shown.
