Rewrite each equation in exponential form. Show your work (2 marks each) ~2 log
=64 I0) 256
The Correct Answer and Explanation is:
To rewrite each equation in exponential form, let’s systematically work through the transformations based on logarithmic definitions.
Given Logarithmic Equations:
- logbA=C\log_b A = C translates to bC=Ab^C = A, where bb is the base, AA is the result, and CC is the exponent.
Using this definition, let’s rewrite the provided equations.
1. log264=6\log_2 64 = 6
The base here is 2, the result is 64, and the exponent is 6.
In exponential form:
26=642^6 = 64
Explanation: The logarithm states that 2 raised to the power of 6 equals 64. This follows directly from the definition of logarithms.
2. log256I0=1\log_{256} I_0 = 1
The base is 256, the result is I0I_0, and the exponent is 1.
In exponential form:
2561=I0256^1 = I_0
Explanation: Since any number raised to the power of 1 is itself, 2561=256256^1 = 256. Here, I0I_0 must equal 256 for the logarithmic equation to hold.
Extended Explanation:
Logarithmic and exponential forms are inverse operations. Logarithms answer the question: “To what power must the base be raised to get a certain number?” By converting logarithmic equations to exponential ones, we express that relationship directly. The base remains unchanged, while the exponent moves to the right-hand side.
In practical applications, this concept appears in sciences like physics and engineering, where exponential relationships govern phenomena such as sound intensity, population growth, and radioactive decay. Recognizing and converting between logarithmic and exponential forms enhances problem-solving capabilities across disciplines.
