To solve the equation [tex]\( 9^{10x - 3} = 7^{6x - 6} \)[/tex], we proceed step-by-step:
1. Take the natural logarithm of both sides:
Start by applying the natural logarithm to both sides of the equation to make use of the properties of logarithms:
[tex]\[
\ln(9^{10x - 3}) = \ln(7^{6x - 6})
\][/tex]
2. Apply the power rule of logarithms:
The power rule states that [tex]\(\ln(a^b) = b \cdot \ln(a)\)[/tex]. Apply this rule to both sides:
[tex]\[
(10x - 3) \cdot \ln(9) = (6x - 6) \cdot \ln(7)
\][/tex]
3. Expand and simplify:
Distribute the logarithms through the parentheses:
[tex]\[
10x \cdot \ln(9) - 3 \cdot \ln(9) = 6x \cdot \ln(7) - 6 \cdot \ln(7)
\][/tex]
Rewrite more cleanly:
[tex]\[
10x \ln(9) - 3 \ln(9) = 6x \ln(7) - 6 \ln(7)
\][/tex]
4. Group terms involving [tex]\( x \)[/tex] on one side:
Move all terms involving [tex]\( x \)[/tex] to one side and constant terms to the other:
[tex]\[
10x \ln(9) - 6x \ln(7) = -6 \ln(7) + 3 \ln(9)
\][/tex]
5. Factor out [tex]\( x \)[/tex] from the left-hand side:
Factor [tex]\( x \)[/tex] out from the left-hand side terms:
[tex]\[
x (10 \ln(9) - 6 \ln(7)) = 3 \ln(9) - 6 \ln(7)
\][/tex]
6. Solve for [tex]\( x \)[/tex]:
Isolate [tex]\( x \)[/tex] by dividing both sides by [tex]\( (10 \ln(9) - 6 \ln(7)) \)[/tex]:
[tex]\[
x = \frac{3 \ln(9) - 6 \ln(7)}{10 \ln(9) - 6 \ln(7)}
\][/tex]
Now, plug in the numerical values for [tex]\(\ln(9)\)[/tex] and [tex]\(\ln(7)\)[/tex], and simplify the expression. Use a calculator for the evaluations:
[tex]\[
\ln(9) \approx 2.1972 \quad \text{and} \quad \ln(7) \approx 1.9459
\][/tex]
Plug these values into your equation:
[tex]\[
x \approx \frac{3 \times 2.1972 - 6 \times 1.9459}{10 \times 2.1972 - 6 \times 1.9459}
\][/tex]
[tex]\[
x \approx \frac{6.5916 - 11.6754}{21.972 - 11.6754}
\][/tex]
[tex]\[
x \approx \frac{-5.0838}{10.2966}
\][/tex]
[tex]\[
x \approx -0.4937
\][/tex]
Thus, the solution to the equation [tex]\( 9^{10x - 3} = 7^{6x - 6} \)[/tex], rounded to four decimal places, is:
[tex]\[
x = -0.4937
\][/tex]
