Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. log₃x=4

Verified step by step guidance

1. Identify the given logarithmic equation: $\log 3_{}x=4$.
2. Recall the definition of logarithm: $\log {\mathrm{a}}_{}b=\; c$ means $a^c\; =\; b$. Applying this, rewrite the equation as $3^4\; =\; x$.
3. Calculate the value of $3^4$ to find the exact value of $x$. (You can leave it as an expression for now.)
4. Check the domain of the original logarithmic expression: since $log\_3\; x$ is defined only for $x\; >\; 0$, verify that the solution satisfies this condition.
5. If needed, use a calculator to find the decimal approximation of $x$ to two decimal places.

Key Concepts

Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log₃ x = 4 means 3 raised to the power 4 equals x. Understanding this definition allows you to rewrite logarithmic equations in exponential form to solve for the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function log_b(x) includes only positive real numbers (x > 0) because logarithms of zero or negative numbers are undefined. When solving logarithmic equations, it is essential to check that solutions fall within this domain to ensure they are valid.

Exact and Approximate Solutions

Logarithmic equations often yield exact solutions expressed in exponential form. However, when exact values are complicated or irrational, calculators can provide decimal approximations. It is important to present both the exact answer and a rounded decimal approximation when required.