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. log3 x=4

Verified step by step guidance

1. Identify the given equation: \( \log_3 x = 4 \).
2. Rewrite the logarithmic equation in its exponential form: \( x = 3^4 \).
3. Calculate the power: \( 3^4 \) to find the value of \( x \).
4. Verify that the solution is within the domain of the original logarithmic expression, which requires \( x > 0 \).
5. If necessary, use a calculator to find a decimal approximation of the solution.

Key Concepts

Logarithmic Functions

A logarithmic function is the inverse of an exponential function, expressed as log_b(a) = c, which means b^c = a. Understanding logarithms is crucial for solving equations involving them, as they allow us to express relationships between numbers in a multiplicative context. In this case, log3(x) = 4 implies that 3 raised to the power of 4 equals x.

Domain of Logarithmic Functions

The domain of a logarithmic function is the set of all positive real numbers, as logarithms are undefined for zero and negative values. When solving logarithmic equations, it is essential to check that any potential solutions fall within this domain to ensure they are valid. For the equation log3(x) = 4, x must be greater than zero.

Exact and Approximate Solutions

In solving logarithmic equations, an exact solution is typically expressed in terms of logarithms or exponents, while an approximate solution is a numerical value obtained through calculation. For the equation log3(x) = 4, the exact solution is x = 3^4, which equals 81. A calculator can then be used to confirm this value or to find its decimal approximation, if necessary.