Let's solve the equation [tex]\( 17^x = 70 \)[/tex] step-by-step and round the answer to two decimal places.
1. Understand the Equation:
We need to find the value of [tex]\( x \)[/tex] such that [tex]\( 17^x = 70 \)[/tex].
2. Use the Natural Logarithm:
To solve for [tex]\( x \)[/tex], we take the natural logarithm (ln) of both sides of the equation. This step is useful because the natural logarithm allows us to use properties of logarithms to bring the exponent [tex]\( x \)[/tex] down, which makes it easier to solve for [tex]\( x \)[/tex].
[tex]\[
\ln(17^x) = \ln(70)
\][/tex]
3. Apply Logarithm Properties:
We use the power rule of logarithms, which states that [tex]\( \ln(a^b) = b \cdot \ln(a) \)[/tex]. Thus, the equation becomes:
[tex]\[
x \cdot \ln(17) = \ln(70)
\][/tex]
4. Isolate [tex]\( x \)[/tex]:
To isolate [tex]\( x \)[/tex], we divide both sides of the equation by [tex]\( \ln(17) \)[/tex]:
[tex]\[
x = \frac{\ln(70)}{\ln(17)}
\][/tex]
5. Calculate the Natural Logarithms:
Let's find the values for [tex]\( \ln(70) \)[/tex] and [tex]\( \ln(17) \)[/tex]:
[tex]\[
\ln(70) \approx 4.248495242049359
\][/tex]
[tex]\[
\ln(17) \approx 2.833213344056216
\][/tex]
6. Divide the Logarithms:
Now, divide [tex]\( \ln(70) \)[/tex] by [tex]\( \ln(17) \)[/tex]:
[tex]\[
x = \frac{4.248495242049359}{2.833213344056216} \approx 1.499532412891622
\][/tex]
7. Round the Result:
Finally, we round the result to two decimal places:
[tex]\[
x \approx 1.50
\][/tex]
Thus, the solution to the equation [tex]\( 17^x = 70 \)[/tex], rounded to two decimal places, is [tex]\( x = 1.50 \)[/tex]. The correct answer is 1.50.
