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. log₄(x+5)=3

Verified step by step guidance

1. Identify the given logarithmic equation: \( \log_4(x+5) = 3 \). This means the logarithm base 4 of \( x+5 \) equals 3.
2. Rewrite the logarithmic equation in its equivalent exponential form using the definition of logarithms: \( \log_b(a) = c \) is equivalent to \( a = b^c \). So, \( x + 5 = 4^3 \).
3. Calculate the right side exponentiation without finalizing the solution: \( 4^3 \) means 4 multiplied by itself 3 times, which is \( 4 \times 4 \times 4 \).
4. Solve for \( x \) by isolating it on one side: \( x = 4^3 - 5 \).
5. Check the domain restriction for the logarithm: since \( \log_4(x+5) \) requires \( x+5 > 0 \), ensure that the solution satisfies \( x > -5 \). If it does, the solution is valid.

Key Concepts

Definition and Properties of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log4(x+5) = 3 means 4 raised to the power 3 equals x+5. Understanding this definition allows you to rewrite logarithmic equations as exponential equations to solve for the variable.

Domain of Logarithmic Functions

The domain of a logarithmic function includes all values for which the argument (inside the log) is positive. For log4(x+5), the expression x+5 must be greater than zero, so x > -5. Checking the domain ensures that solutions are valid and prevents taking logarithms of non-positive numbers.

Solving Logarithmic Equations

To solve log equations like log4(x+5) = 3, convert the logarithmic form to exponential form: x+5 = 4^3. Then solve the resulting equation for x. After finding solutions, verify they lie within the domain to reject any extraneous answers.