Solve for $$ a $$ and $$ b $$ in the following system of equations: $$ 2^{3 a}=7^{b} $$ $$ 3^{a-1}=13 $$

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UpStudy AI · Accepted Answer

We are given the system
$$
2^{3a} = 7^b
$$
$$
3^{a-1} = 13
$$

**Step 1. Solve for $$ a $$ using $$ 3^{a-1} = 13 $$:**

Take the natural logarithm on both sides:
$$
\ln\left(3^{a-1}\right) = \ln(13)
$$
Using the logarithm rule $$ \ln(x^y) = y \ln(x) $$, we have:
$$
(a-1) \ln(3) = \ln(13)
$$
Solve for $$ a $$:
$$
a-1 = \frac{\ln(13)}{\ln(3)}
$$
$$
a = 1 + \frac{\ln(13)}{\ln(3)}
$$

**Step 2. Solve for $$ b $$ using $$ 2^{3a} = 7^b $$:**

Taking the natural logarithm on both sides:
$$
\ln\left(2^{3a}\right) = \ln\left(7^b\right)
$$
Using the logarithm rule, we get:
$$
3a \ln(2) = b \ln(7)
$$
Solve for $$ b $$:
$$
b = \frac{3a \ln(2)}{\ln(7)}
$$

Substitute the expression we obtained for $$ a $$:
$$
b = \frac{3 \ln(2)}{\ln(7)} \left(1 + \frac{\ln(13)}{\ln(3)}\right)
$$

**Final Answers:**

$$
a = 1 + \frac{\ln(13)}{\ln(3)}
$$
$$
b = \frac{3 \ln(2)}{\ln(7)} \left(1 + \frac{\ln(13)}{\ln(3)}\right)
$$
$$ a = 1 + \frac{\ln(13)}{\ln(3)} $$ $$ b = \frac{3 \ln(2)}{\ln(7)} \left(1 + \frac{\ln(13)}{\ln(3)}\right) $$

Solve for and in the following system of equations:

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