Question12. Why must the base of a logarithm be positive? [3 marks]
Studdy Solution
STEP 1
1. Logarithms are mathematical functions that are the inverse of exponentiation.
2. The base of a logarithm is a positive real number, not equal to 1.
STEP 2
1. Explain the definition and properties of logarithms.
2. Discuss the implications of having a non-positive base.
3. Conclude why the base must be positive.
STEP 3
A logarithm is defined as the exponent such that .
The base must be a positive real number because exponentiation with a positive base is well-defined and results in real numbers for all real exponents.
STEP 4
If the base were zero or negative, the function would not be well-defined for all real .
For example, a zero base would make for any positive , and undefined for zero or negative .
A negative base would result in complex numbers for non-integer exponents, which is not suitable for the real-valued logarithm function.
STEP 5
The base of a logarithm must be positive to ensure that the logarithmic function is well-defined and results in real numbers for all positive .
This ensures the function is continuous and behaves predictably across its domain.
The base of a logarithm must be positive to maintain the integrity and definition of the logarithmic function in real numbers.
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