Consider the equation $\newline$ $11 \cdot 10^{5 t}=20 \text {. }$ $\newline$ Solve the equation for $t$ . Express the solution as a logarithm in base $10$ . $\newline$ $t=$ $\newline$ Approximate the value of $t$ . Round your answer to the nearest thousandth. $\newline$ $t \approx$

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Algebra 2

Change of base formula

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Q. Consider the equation

\newline

11 \cdot 10^{5 t}=20 \text {. }

\newline

Solve the equation for

t

. Express the solution as a logarithm in base

10

\newline

t=

\newline

Approximate the value of

t

. Round your answer to the nearest thousandth.

\newline

t \approx

Isolate exponential term: Isolate the exponential term. $\newline$ To solve for $t$ , we first need to isolate the term with the exponent. We do this by dividing both sides of the equation by $11$ . $\newline$ $11 \cdot 10^{5t} = 20$ $\newline$ $10^{5t} = \frac{20}{11}$
Apply logarithm to both sides: Apply the logarithm to both sides of the equation. $\newline$ To solve for the exponent, we apply the logarithm to both sides of the equation. We will use the common logarithm (base $10$ ). $\newline$ $\log(10^{5t}) = \log\left(\frac{20}{11}\right)$
Use property of logarithms: Use the property of logarithms to bring down the exponent. $\newline$ The property of logarithms that we use here is $\log(b^x) = x \cdot \log(b)$ . We apply this property to the left side of the equation. $\newline$ $5t \cdot \log(10) = \log\left(\frac{20}{11}\right)$
Simplify left side: Simplify the left side of the equation. $\newline$ Since $\log(10)$ is equal to $1$ , the equation simplifies to: $\newline$ $5t = \log\left(\frac{20}{11}\right)$
Solve for t: Solve for t. $\newline$ To solve for t, we divide both sides of the equation by $5$ . $\newline$ $t = \frac{\log(\frac{20}{11})}{5}$
Approximate value of t: Approximate the value of t. $\newline$ Using a calculator, we can find the approximate value of t by evaluating the logarithm and dividing by $5$ . $\newline$ $t \approx \frac{\log(\frac{20}{11})}{5}$ $\newline$ $t \approx \frac{\log(1.81818181818)}{5}$ $\newline$ $t \approx \frac{0.2596373105}{5}$ $\newline$ $t \approx 0.0519274621$ $\newline$ Rounded to the nearest thousandth, $t \approx 0.052$