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Of all eligible voters in your county,
70% are currently registered to vote.
A recent poll predicts that the relationship between
V, the percentage of all eligible voters who are registered to vote, and
t, the number of years from now will be modeled by the following equation.

V=100-30*e^(-0.04 t)
In how many years will
80% of all eligible voters in your county be registered to vote?
Give an exact answer expressed as a natural logarithm.
years

Of all eligible voters in your county, $70 \%$ are currently registered to vote. $\newline$ A recent poll predicts that the relationship between $V$ , the percentage of all eligible voters who are registered to vote, and $t$ , the number of years from now will be modeled by the following equation. $\newline$ $V=100-30 \cdot e^{-0.04 t}$ $\newline$ In how many years will $80 \%$ of all eligible voters in your county be registered to vote? $\newline$ Give an exact answer expressed as a natural logarithm. $\newline$ years

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Convert between exponential and logarithmic form: all bases

Full solution

Q. Of all eligible voters in your county,

70 \%

are currently registered to vote.

\newline

A recent poll predicts that the relationship between

V

, the percentage of all eligible voters who are registered to vote, and

t

, the number of years from now will be modeled by the following equation.

\newline

V=100-30 \cdot e^{-0.04 t}

\newline

In how many years will

80 \%

of all eligible voters in your county be registered to vote?

\newline

Give an exact answer expressed as a natural logarithm.

\newline

years

Set up equation with $V$ : Set up the equation with $V$ equal to $80$ . We are given the equation $V = 100 - 30\cdot e^{(-0.04t)}$ and we want to find the value of $t$ when $V$ is $80$ . So, we set $V$ to $80$ and solve for $t$ . $V$ $0$
Subtract to isolate exponential term: Subtract $100$ from both sides of the equation. $\newline$ $80 - 100 = 100 - 30e^{-0.04t} - 100$ $\newline$ $-20 = -30e^{-0.04t}$
Divide sides by $-30$ : Divide both sides by $-30$ to isolate the exponential term. $\newline$ $-20 / -30 = (-30*e^{(-0.04t)}) / -30$ $\newline$ $\frac{2}{3} = e^{(-0.04t)}$
Take natural logarithm: Take the natural logarithm of both sides to solve for $t$ . $\ln(\frac{2}{3}) = \ln(e^{-0.04t})$
Simplify right side: Use the property of logarithms that $\ln(e^x) = x$ to simplify the right side of the equation. $\ln(\frac{2}{3}) = -0.04t$
Divide sides by $-0.04$ : Divide both sides by $-0.04$ to solve for $t$ . $t = \frac{\ln(\frac{2}{3})}{-0.04}$

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