Final answer:
To rewrite a logarithmic equation without logs, transform it into the exponential form, where a logarithmic function and its base are inverses, such as 'log_b(x) = y' becoming 'b^y = x'. Polynomial forms express logarithms as equivalent equations without logs, like 'log(x-3) = 2' being '(x-3)^2 = 100'. Radical forms apply when equations involve exponential functions with fractional exponents, and trigonometric forms are used less frequently.
Step-by-step explanation:
To rewrite a logarithmic equation in various forms without using logs, you can use exponential, polynomial, radical, or trigonometric forms depending on the initial equation. For the exponential form, logarithms and exponentials are inverse functions, for example:
logb(x) = y is equivalent to by = x.
For polynomial form, some logarithmic equations can be written as polynomials when you exponentiate them to remove the logarithm. For example, you can express the logarithmic equation log(x-3) = 2 as the polynomial equation (x-3)2 = 100.
The radical form typically involves exponential equations with fractional exponents. For example, log4(x) = 1/2 can be rewritten as x = 41/2, which simplifies to the radical form x = −4.
In trigonometric form, logarithms are rarely used, but an example would be taking the inverse trigonometric function of a logarithmic value, like so: log10(sin(x)) = -1 could be rewritten as sin(x) = 10-1 or sin(x) = 1/10.