Consider the equation [tex]$5 \cdot 10^{\frac{z}{4}}=32$[/tex].
Solve the equation for [tex]$z$[/tex]. Express the solution as a logarithm in base-10.
[tex]$z=$[/tex]
Approximate the value of [tex]$z$[/tex]. Round your answer to the nearest thousandth.
[tex]$z \approx$[/tex]
Answer (2)
Divide both sides of the equation by 5 to get 1 0 4 z = 6.4 .
Take the base-10 logarithm of both sides: 4 z = lo g 10 ( 6.4 ) .
Multiply both sides by 4: z = 4 lo g 10 ( 6.4 ) .
Approximate the value of z to the nearest thousandth: z ≈ 3.225 .
Explanation
Understanding the Problem We are given the equation 5 c d o t 1 0 4 z = 32 . Our goal is to solve for z , express the solution as a base-10 logarithm, and approximate the value of z to the nearest thousandth.
Isolating the Exponential Term First, we divide both sides of the equation by 5 to isolate the exponential term: 5 5 c d o t 1 0 4 z = 5 32 1 0 4 z = 5 32 1 0 4 z = 6.4
Taking the Base-10 Logarithm Next, we take the base-10 logarithm of both sides of the equation: lo g 10 ( 1 0 4 z ) = lo g 10 ( 6.4 )
Simplifying the Logarithm Using the property of logarithms that lo g b ( b x ) = x , we simplify the left side of the equation: 4 z = lo g 10 ( 6.4 )
Solving for z Now, we multiply both sides of the equation by 4 to solve for z :
z = 4 c d o t lo g 10 ( 6.4 )
Approximating the Logarithm To approximate the value of z , we first find the value of lo g 10 ( 6.4 ) . The result of this operation is approximately 0.8061799739838872. lo g 10 ( 6.4 ) ≈ 0.8061799739838872
Calculating z Now, we multiply this value by 4: z ≈ 4 c d o t 0.8061799739838872 ≈ 3.2247198959355488
Rounding to the Nearest Thousandth Finally, we round the value of z to the nearest thousandth: z ≈ 3.225
Final Answer Therefore, the solution for z is 4 lo g 10 ( 5 32 ) and the approximate value of z rounded to the nearest thousandth is 3.225.
Examples
Logarithms are incredibly useful in many real-world scenarios, especially when dealing with exponential growth or decay. For instance, in finance, you can use logarithms to calculate the time it takes for an investment to double at a certain interest rate. Similarly, in environmental science, logarithms help in measuring the intensity of earthquakes using the Richter scale or determining the acidity (pH) of a solution. Understanding logarithms allows us to make informed decisions and predictions in various fields, from personal finance to scientific research.
To solve the equation 5 ⋅ 1 0 4 z = 32 , we isolate the exponential part and take the logarithm, leading to z = 4 lo g 10 ( 6.4 ) . The approximate value of z rounds to 3.225. Thus, z is expressed as a logarithm and approximated numerically.
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