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Consider the equation

5*10^((z)/(4))=32". "
Solve the equation for
z. Express the solution as a logarithm in base10.

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

z~~

Consider the equation $\newline$ $5 \cdot 10^{\frac{z}{4}}=32 \text {. }$ $\newline$ Solve the equation for $z$ . Express the solution as a logarithm in base $10$ . $\newline$ $z=$ $\newline$ Approximate the value of $z$ . Round your answer to the nearest thousandth. $\newline$ $z \approx$

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Compare linear and exponential growth

Full solution

Q. Consider the equation

\newline

5 \cdot 10^{\frac{z}{4}}=32 \text {. }

\newline

Solve the equation for

z

. Express the solution as a logarithm in base

10

.

\newline

z=

\newline

Approximate the value of

z

. Round your answer to the nearest thousandth.

\newline

z \approx

Write and Identify Equation: Write down the given equation and identify the goal. $\newline$ The given equation is $5\cdot10^{\frac{z}{4}} = 32$ . We need to solve for $z$ .
Isolate Exponential Term: Isolate the exponential term. $\newline$ To isolate the exponential term, divide both sides of the equation by $5$ . $\newline$ $10^{z/4} = \frac{32}{5}$
Convert to Logarithmic Form: Convert the equation to logarithmic form. $\newline$ To solve for $z$ , we can take the logarithm of both sides of the equation. We will use the base $10$ logarithm since our exponential base is $10$ . $\newline$ $\log(10^{z/4}) = \log(\frac{32}{5})$
Apply Power Rule of Logarithms: Apply the power rule of logarithms. $\newline$ The power rule of logarithms states that $\log(a^b) = b\cdot\log(a)$ . We apply this rule to the left side of the equation. $\newline$ $\frac{z}{4}\cdot\log(10) = \log\left(\frac{32}{5}\right)$
Simplify Left Side of Equation: Simplify the left side of the equation. $\newline$ Since $\log(10)$ is equal to $1$ , the equation simplifies to: $\newline$ $\frac{z}{4} = \log\left(\frac{32}{5}\right)$
Solve for z: Solve for z. $\newline$ To solve for z, multiply both sides of the equation by $4$ . $\newline$ $z = 4\log\left(\frac{32}{5}\right)$
Calculate Approximate Value of z: Calculate the approximate value of z. $\newline$ Using a calculator, find the value of $\log(\frac{32}{5})$ and then multiply by $4$ to get $z$ . $\newline$ $z \approx 4\cdot\log(\frac{32}{5})$ $\newline$ $z \approx 4\cdot0.8062$ (rounded to four decimal places) $\newline$ $z \approx 3.2248$

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