Problem 53 - Algebra and Trigonometry

Question

Solve each logarithmic equation. 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. $$ \log _{4}(x+5)=3 $$

Step-by-Step Solution

Answer

The solution to the equation \( \log _{4}(x+5) = 3 \) is \( x = 59 \).

Step 1: Apply the Definition of Logarithms

In this step, change the logarithmic equation to an exponential equation. Hence, \( \log _{4}(x+5) = 3 \) can be written as \( 4^3 = x + 5 \).

Step 2: Evaluate Exponential and Solve for x

Calculate the value of \( 4^3 \) which is 64. Therefore the equation becomes \( 64 = x + 5 \). Now, solve for \( x \) by subtracting 5 from both sides, the equation becomes \( x = 64 - 5 \).

Step 3: Calculate the Value of x

Subtract 5 from 64 to get the final value of \( x \) which is 59.

Step 4: Validate the Solution

Substitute \( x = 59 \) back into the original logarithmic expression \( \log _{4}(x+5) \). If a valid log (greater than zero), then 59 is a valid solution.

Key Concepts

Definition of Logarithms

Logarithms are a way to express a number as an exponent. In simpler terms, a logarithm tells you which power you need to raise a base number to, in order to reach another number. For example, in our equation \( \log_{4}(x+5) = 3 \), we read this as 'the logarithm base 4 of \( x+5 \) equals 3'. This means that we need to find the number which, when 4 is raised to that power (in this case, 3), equals \( x+5 \). A useful way to remember this is through the question: '4 raised to what power results in another value?' This is integral to solving logarithmic equations.

To convert a logarithmic expression to an exponential equation, you rearrange the equation like so: if \( \log_b(a) = c \), then \( b^c = a \). This conversion helps us solve for the unknown in problems involving logarithms. This foundational understanding is crucial for algebraic calculations and for moving easily between different forms of mathematical expressions.

Exponential Equations

Exponential equations are equations where variables appear as exponents. They are closely connected with logarithms, as the logarithm is essentially the inverse operation to exponentiation. Taking our exercise, by turning the logarithmic equation \( \log_{4}(x+5) = 3 \) into an exponential equation, we found that \( 4^3 = x+5 \).

Exponential equations are powerful. Solving them typically involves isolating the exponential part on one side of the equation, calculating the power, and then solving for the variable. Here, \( 4^3 \) was calculated to be 64, which implies \( x+5 = 64 \).

Breaking it down, once you find the exponential value, adjust the rest of the equation to isolate the variable. In this example, to solve for \( x \), we subtracted 5 from both sides of the equation, resulting in \( x = 59 \). This methodical approach allows for error-free calculation.

Solution Validation

Validating a mathematical solution is essential to ensure the correctness of the answer. After finding a solution, especially in equations involving logarithms, verify that the result is meaningful and satisfies original conditions. For the problem \( \log_{4}(x+5) = 3 \), after finding \( x = 59 \), we substitute this back into the logarithmic equation to check if it holds true.

Substitution helps confirm that the solution lies within the domain, which in the case of logarithms, demands the value inside the logarithm to be positive. If \( x+5 \) must remain positive for the expression to be valid, substituting \( x = 59 \) results in \( \log_{4}(64) \). Since 64 is positive, our solution is validated as correct.

This final check ensures not only numerical accuracy but also checks logical compliance with the properties of logarithims, making sure the solution is complete and necessary for obtaining a fully correct result.