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. Start by understanding the equation: \( \log_4(x+5) = 3 \). This is a logarithmic equation where the base is 4.
2. To solve for \( x \), convert the logarithmic equation to its exponential form. Recall that \( \log_b(a) = c \) is equivalent to \( b^c = a \).
3. Apply this conversion: \( 4^3 = x + 5 \).
4. Calculate \( 4^3 \) to simplify the equation: \( 64 = x + 5 \).
5. Isolate \( x \) by subtracting 5 from both sides: \( x = 64 - 5 \).

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The equation log_b(a) = c means that b raised to the power of c equals a (b^c = a). Understanding this relationship is crucial for solving logarithmic equations, as it allows us to rewrite the logarithmic expression in exponential form.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. For the expression log_b(x), x must be greater than zero (x > 0) because logarithms of non-positive numbers are undefined. When solving logarithmic equations, it is essential to check that any solutions fall within this domain to ensure they are valid.

Exact and Approximate Solutions

In solving logarithmic equations, it is often necessary to provide both exact solutions and decimal approximations. The exact solution is typically expressed in terms of logarithms or algebraic expressions, while the decimal approximation is obtained using a calculator to provide a numerical value, rounded to a specified number of decimal places, which aids in practical applications.