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Chapter 3: Problem 33

Solve the exponential equation algebraically. Approximate the result to three decimal places. $$4\left(3^{x}\right)=20$$

Short Answer

Expert verified
The solution to the equation is approximately \(x = 1.465\).

Step by step solution

01

Rewrite in exponential form

Rewrite the given equation \(4 \times 3^{x} = 20\) in exponential form by isolating the exponential term, which means dividing by 4 on both sides of the equation. This will give: \(3^{x} = \frac{20}{4}\), which simplifies to \(3^{x} = 5\)
02

Convert to logarithmic form

Convert the equation from exponential form to logarithmic form. This changes the base of the equation and allows for easier manipulation of the terms. This gives: \(\log_{3}5 = x\)
03

Solve for x

To solve for x, use the properties of logarithms to evaluate the expression. Since we do not have an exact value for \(\log_{3}5\), we can use a calculator to find the approximate value, which can then be rounded to 3 decimal places. This gives approximately \(x = 1.465\).

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Most popular questions from this chapter

Find the exponential model \(y=a e^{b x}\) that fits the points shown in the graph or table. $$ \begin{array}{|l|l|l|} \hline x & 0 & 4 \\ \hline y & 5 & 1 \\ \hline \end{array} $$

Determine the principal \(P\) that must be invested at rate \(r\), compounded monthly, so that $$\$ 500,000$$ will be available for retirement in \(t\) years. $$r=5 \%, t=10$$

Use the acidity model given by \(\mathrm{pH}=-\log \left[\mathrm{H}^{+}\right],\) where acidity \((\mathrm{pH})\) is a measure of the hydrogen ion concentration \(\left[\mathrm{H}^{+}\right]\) (measured in moles of hydrogen per liter) of a solution. Find the \(\mathrm{pH}\) if \(\left[\mathrm{H}^{+}\right]=1.13 \times 10^{-5}\).

Solve the logarithmic equation algebraically. Approximate the result to three decimal places. $$6 \log _{3}(0.5 x)=11$$

Solve the equation algebraically. Round the result to three decimal places. Verify your answer using a graphing utility. $$-x e^{-x}+e^{-x}=0$$
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