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Consider the equation

-3*10^(2t)=-28". "
Solve the equation for
t. Express the solution as a logarithm in base10.

t=
Approximate the value of
t. Round your answer to the nearest thousandth.

t~~

Consider the equation $\newline$ $-3 \cdot 10^{2 t}=-28 \text {. }$ $\newline$ 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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Change of base formula

Full solution

Q. Consider the equation

\newline

-3 \cdot 10^{2 t}=-28 \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 exponential term $10^{2t}$ . We can do this by dividing both sides of the equation by $-3$ . $\newline$ $-3\cdot10^{2t} = -28$ $\newline$ $10^{2t} = -28 / -3$ $\newline$ $10^{2t} = 28 / 3$
Take common logarithm: Take the common logarithm of both sides. $\newline$ To solve for the exponent, we take the logarithm of both sides of the equation. We use the common logarithm (base $10$ ). $\newline$ $\log(10^{2t}) = \log(\frac{28}{3})$
Apply power rule of logarithms: Apply the power rule of logarithms. $\newline$ The power rule of logarithms states that $\log(b^x) = x\log(b)$ . We apply this rule to the left side of the equation. $\newline$ $2t \cdot \log(10) = \log\left(\frac{28}{3}\right)$ $\newline$ Since $\log(10)$ is $1$ , the equation simplifies to: $\newline$ $2t = \log\left(\frac{28}{3}\right)$
Solve for t: Solve for t. $\newline$ To solve for t, we divide both sides of the equation by $2$ . $\newline$ $t = \log(\frac{28}{3}) / 2$
Approximate value of $t$ : Approximate the value of $t$ . We can now use a calculator to find the approximate value of $t$ . $t \approx \log(\frac{28}{3}) / 2$ $t \approx \log(9.3333\ldots) / 2$ $t \approx 0.9698 / 2$ $t \approx 0.4849$

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