Answers:
When we evaluate a logarithm, we are finding the exponent, or power x, that the base b, needs to be raised so that it equals the argument m. The power is also known as the exponent.
[tex]5^2 = 25 \to \log_5(25) = 2[/tex]
The value of b must be positive and not equal to 1
The value of m must be positive
If 0 < m < 1, then x < 0
A logarithmic equation is an equation with a variable that includes one or more logarithms.
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Explanation:
Logarithms, or log for short, basically undo what exponents do.
When going from [tex]5^2 = 25[/tex] to [tex]\log_5(25) = 2[/tex], we have isolated the exponent.
More generally, we have [tex]b^x = m[/tex] turn into [tex]\log_b(m) = x[/tex]
When using the change of base formula, notice how
[tex]\log_b(m) = \frac{\log(m)}{\log(b)}[/tex]
If b = 1, then log(b) = log(1) = 0, meaning we have a division by zero error. So this is why [tex]b \ne 1[/tex]
We need b > 0 as well because the domain of y = log(x) is the set of positive real numbers. So this is why m > 0 also.
