Question18. The approximate value of $x$ in the equation $17^{x-1}=12^{x+3}$ is A. 10.6 B. 12.6 C. 29.5 D. 31.5

Studdy Solution

STEP 1

What is this asking? We need to find the value of $x$ that makes $17^{x-1}$ roughly equal to $12^{x+3}$ . Watch out! Don't get intimidated by exponents!
We'll use logarithms to bring them down.

STEP 2

1. Take the logarithm of both sides
2. Expand using logarithm rules
3. Isolate $x$
4. Calculate the approximate value of $x$

STEP 3

Let's apply the logarithm to both sides of our equation $17^{x-1} = 12^{x+3}$ .
We'll use the common logarithm (base 10), but any logarithm would work!
This gives us $\log(17^{x-1}) = \log(12^{x+3})$ .
Why are we doing this?
Because logarithms have a special property that lets us bring down exponents, which is exactly what we want to do!

STEP 4

Remember the power rule of logarithms: $\log(a^b) = b \cdot \log(a)$ .
Let's use this rule to bring down those exponents!
We get $(x-1) \cdot \log(17) = (x+3) \cdot \log(12)$ .
Now, our equation looks much more manageable!

STEP 5

Now, let's distribute the logarithms on both sides: $x \cdot \log(17) - \log(17) = x \cdot \log(12) + 3 \cdot \log(12)$ .
We're getting closer to isolating $x$ !

STEP 6

Let's move all terms with $x$ to one side and all the constant terms to the other.
We'll subtract $x \cdot \log(12)$ from both sides and add $\log(17)$ to both sides.
This gives us $x \cdot \log(17) - x \cdot \log(12) = 3 \cdot \log(12) + \log(17)$ .

STEP 7

We can factor out $x$ from the left side: $x \cdot (\log(17) - \log(12)) = 3 \cdot \log(12) + \log(17)$ .
Almost there!

STEP 8

Finally, we can divide both sides by $(\log(17) - \log(12))$ to isolate $x$ : $x = \frac{3 \cdot \log(12) + \log(17)}{\log(17) - \log(12)}$ .

STEP 9

We know that $\log(17) \approx 1.23$ and $\log(12) \approx 1.08$ .
Substituting these values into our equation, we get $x \approx \frac{3 \cdot 1.08 + 1.23}{1.23 - 1.08} = \frac{3.24 + 1.23}{0.15} = \frac{4.47}{0.15}$ .

STEP 10

Now, we just need to perform the final division: $x \approx \frac{4.47}{0.15} \approx 29.8$ .
This is closest to the answer choice C.
29.5.

STEP 11

The approximate value of $x$ is 29.5, so the answer is C.

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