Solve the following equations: 8^(t)=6

Recognize Problem: Recognize that the equation 8^t = 6 cannot be solved using simple exponent rules since 6 is not a power of 8.Apply Logarithms: Apply logarithms to both sides of the equation to solve for t. We can use the natural logarithm (ln) for this purpose. ln(8^t) = ln(6)Use Power Rule: Use the power rule of logarithms which states that ln(a^b) = b * ln(a) to simplify the left side of the equation. t * ln(8) = ln(6)Solve for t: Solve for t by dividing both sides of the equation by ln(8). t = ln(6) / ln(8)Calculate Value: Calculate the value of t using a calculator. t ≈ ln(6) / ln(8) ≈ 0.7737Check Calculation: Check the calculation for any mathematical errors. Using a calculator, we verify that ln(6) / ln(8) indeed equals approximately 0.7737.

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Solve the following equations: $\newline$ $1$ ) $8^{t}=6$

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Algebra 2

Solve exponential equations by rewriting the base

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Q. Solve the following equations:

\newline

1

)

8^{t}=6

Recognize Problem: Recognize that the equation $8^t = 6$ cannot be solved using simple exponent rules since $6$ is not a power of $8$ .
Apply Logarithms: Apply logarithms to both sides of the equation to solve for $t$ . We can use the natural logarithm ( $\ln$ ) for this purpose. $\newline$ $\ln(8^t) = \ln(6)$
Use Power Rule: Use the power rule of logarithms which states that $\ln(a^b) = b \cdot \ln(a)$ to simplify the left side of the equation. $\newline$ $t \cdot \ln(8) = \ln(6)$
Solve for t: Solve for t by dividing both sides of the equation by $\ln(8)$ . $t = \frac{\ln(6)}{\ln(8)}$
Calculate Value: Calculate the value of $t$ using a calculator. $t \approx \frac{\ln(6)}{\ln(8)} \approx 0.7737$
Check Calculation: Check the calculation for any mathematical errors. $\newline$ Using a calculator, we verify that $\ln(6) / \ln(8)$ indeed equals approximately $0.7737$ .