Solve the following for r 8(10^(3r))=12

Write Equation: Write down the given equation. We are given the equation 8(10^(3r)) = 12.Divide and Isolate: Divide both sides of the equation by 8 to isolate the term with the variable r. (10^(3r)) = 12 / 8 (10^(3r)) = 1.5Take Logarithm: Take the logarithm of both sides of the equation to solve for the exponent. We use the common logarithm (base 10) since the base of the exponent is 10. log(10^(3r)) = log(1.5)Apply Power Rule: Apply the power rule of logarithms, which states that log(a^b) = b*log(a), to the left side of the equation. 3r * log(10) = log(1.5) Since log(10) is 1, this simplifies to: 3r = log(1.5)Solve for r: Divide both sides of the equation by 3 to solve for r. r = log(1.5) / 3Calculate Value of r: Calculate the value of r using a calculator. r ≈ log(1.5) / 3 r ≈ 0.176 / 3 r ≈ 0.0587

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Math Problems

Grade 8

Describe the graph of a linear equation

Full solution

Q. Solve the following for `r`

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8(10^{3r})=12

Write Equation: Write down the given equation. $\newline$ We are given the equation $8(10^{3r}) = 12$ .
Divide and Isolate: Divide both sides of the equation by $8$ to isolate the term with the variable $r$ . $\newline$ $(10^{3r}) = \frac{12}{8}$ $\newline$ $(10^{3r}) = 1.5$
Take Logarithm: Take the logarithm of both sides of the equation to solve for the exponent. $\newline$ We use the common logarithm (base $10$ ) since the base of the exponent is $10$ . $\newline$ $\log(10^{3r}) = \log(1.5)$
Apply Power Rule: Apply the power rule of logarithms, which states that $\log(a^b) = b\log(a)$ , to the left side of the equation. $\newline$ $3r \cdot \log(10) = \log(1.5)$ $\newline$ Since $\log(10)$ is $1$ , this simplifies to: $\newline$ $3r = \log(1.5)$
Solve for $r$ : Divide both sides of the equation by $3$ to solve for $r$ . $r = \frac{\log(1.5)}{3}$
Calculate Value of r: Calculate the value of r using a calculator. $\newline$ $r \approx \log(1.5) / 3$ $\newline$ $r \approx 0.176 / 3$ $\newline$ $r \approx 0.0587$