Consider the equation [tex]0.75 \cdot 10^{\frac{w}{3}}=30[/tex].
Solve the equation for [tex]w[/tex]. Express the solution as a logarithm in base-10.
[tex]w=\square[/tex]
Approximate the value of [tex]w[/tex]. Round your answer to the nearest thousandth.
[tex]w \approx \square[/tex]
Answer (2)
Divide both sides of the equation by 0.75 to isolate the exponential term: 1 0 3 w = 40 .
Take the base-10 logarithm of both sides: 3 w = lo g 10 ( 40 ) .
Multiply both sides by 3 to solve for w : w = 3 lo g 10 ( 40 ) .
Approximate the value of w to the nearest thousandth: w ≈ 4.806 .
Explanation
Isolating the Exponential Term We are given the equation 0.75 c d o t 1 0 3 w = 30 and we want to solve for w . First, we will isolate the exponential term.
Dividing by 0.75 Divide both sides of the equation by 0.75 :
0.75 0.75 c d o t 1 0 3 w = 0.75 30 1 0 3 w = 40
Taking the Base-10 Logarithm Now, we take the base-10 logarithm of both sides of the equation: lo g 10 ( 1 0 3 w ) = lo g 10 ( 40 )
Simplifying the Logarithm Using the property of logarithms that lo g b ( b x ) = x , we have: 3 w = lo g 10 ( 40 )
Solving for w Multiply both sides of the equation by 3 to solve for w :
w = 3 c d o t lo g 10 ( 40 )
Approximating the Value of w Now we need to approximate the value of w . We know that w = 3 c d o t lo g 10 ( 40 ) . Using a calculator, we find that lo g 10 ( 40 ) ≈ 1.60206 . Therefore, w ≈ 3 c d o t 1.60206 = 4.80618 Rounding to the nearest thousandth, we get w ≈ 4.806 .
Final Answer Therefore, the solution for w is w = 3 lo g 10 ( 40 ) , and the approximate value of w is 4.806 .
Examples
Logarithms are incredibly useful in many real-world situations, especially when dealing with exponential growth or decay. For example, calculating the time it takes for an investment to double at a certain interest rate involves solving an exponential equation using logarithms. Similarly, in environmental science, logarithms are used to model the decay of radioactive substances or to measure the intensity of earthquakes using the Richter scale. Understanding how to solve logarithmic equations allows us to make informed decisions in finance, science, and many other fields where exponential relationships are prevalent.
The solution for w is given by w = 3 lo g 10 ( 40 ) . After calculating, the approximate value is w ≈ 4.806 .
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