Convert the logarithm to exponential form: [tex]$\log _4(256)=4$[/tex].
A. [tex]$256^4=4$[/tex]
B. [tex]$44=\log _4(256)$[/tex].
C. [tex]$4^{256}=4$[/tex].
D. [tex]$4^4=256$[/tex]
By converting to an exponential expression solve [tex]$\log_4(x+5)=4$[/tex]
Answer (1)
Convert the logarithmic equation lo g 4 ( 256 ) = 4 to exponential form: 4 4 = 256 .
Convert the logarithmic equation lo g a ( x + 5 ) = 4 to exponential form: a 4 = x + 5 .
Isolate x in the equation a 4 = x + 5 : x = a 4 − 5 .
The solution for x is x = a 4 − 5 .
Explanation
Problem Analysis First, let's analyze the given information. We have two tasks: first, to convert the logarithmic equation lo g 4 ( 256 ) = 4 to exponential form and verify its correctness. Second, to solve the equation lo g a ( x + 5 ) = 4 for x .
Converting to Exponential Form For the first task, the logarithmic equation lo g 4 ( 256 ) = 4 means that 4 raised to the power of 4 equals 256. The exponential form is therefore 4 4 = 256 . We can verify this: 4 4 = 4 × 4 × 4 × 4 = 16 × 16 = 256 , which is correct.
Solving for x For the second task, we are given the equation lo g a ( x + 5 ) = 4 . To solve for x , we convert this logarithmic equation to exponential form. The base is a , the exponent is 4, and the result is x + 5 . Therefore, the exponential form is a 4 = x + 5 .
Isolating x Now, we isolate x by subtracting 5 from both sides of the equation: x = a 4 − 5 . This is the solution for x in terms of a .
Final Answer Therefore, the exponential form of lo g 4 ( 256 ) = 4 is 4 4 = 256 , and the solution to lo g a ( x + 5 ) = 4 is x = a 4 − 5 .
Examples
Logarithmic equations are used in many real-world applications, such as calculating the magnitude of earthquakes on the Richter scale, measuring the acidity or alkalinity (pH) of a solution, and modeling population growth or decay. For example, if we know the intensity of an earthquake is 1000 times greater than the smallest detectable wave, we can use logarithms to find its magnitude on the Richter scale: R = lo g 10 ( 1000 ) = 3 . Similarly, in finance, logarithmic scales are used to analyze investment growth and decay over time.
