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Consider the equation
10^((2z)/(3))=15.
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 $10^{\frac{2 z}{3}}=15$ . $\newline$ 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

10^{\frac{2 z}{3}}=15

.

\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 Equation: Write down the given equation. $\newline$ We have the equation $10^{\left(\frac{2z}{3}\right)} = 15$ .
Take Logarithm: Take the logarithm of both sides of the equation to solve for $z$ . $\newline$ We use the property that if $a^b = c$ , then $\log_a(c) = b$ . $\newline$ Taking the logarithm base $10$ of both sides, we get $\log_{10}(10^{(\frac{2z}{3})}) = \log_{10}(15)$ .
Simplify Left Side: Simplify the left side of the equation using the property of logarithms. $\newline$ The logarithm of a power is the exponent times the logarithm of the base, so $\log_{10}(10^{\frac{2z}{3}})$ simplifies to $\frac{2z}{3}$ . $\newline$ Our equation now is $\frac{2z}{3} = \log_{10}(15)$ .
Solve for z: Solve for z. $\newline$ To isolate z, we multiply both sides of the equation by $\frac{3}{2}$ . $\newline$ $z = \left(\frac{3}{2}\right) \cdot \log_{10}(15)$ .
Calculate z: Calculate the value of z using a calculator. $\newline$ Using a calculator, we find that $\log_{10}(15)$ is approximately $1.176$ . $\newline$ So, $z \approx \left(\frac{3}{2}\right) \times 1.176$ .
Perform Multiplication: Perform the multiplication to find the approximate value of $z$ . $\newline$ $z \approx \left(\frac{3}{2}\right) \times 1.176 \approx 1.764$ .

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