Problem 50 - 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. $$ \log _{5} x=3 $$

Step-by-Step Solution

Answer

The solution to the logarithmic equation \(\log _{5} x=3\) is \(x = 125\).

Step 1: Transform the equation

The logarithmic equation can be transformed into an exponential one using the definition of a logarithm. This means \(5^3 = x\)

Step 2: Calculate the value of x

Calculating the value of \(x\), we find that \(x = 5^3\), or 125.

Step 3: Verify domain

Since Logarithmic functions are undefined for negative values and zero, we must ensure that our solution falls within the domain. 125 is a positive number so it falls within the domain of the original logarithmic expression

Step 4: Decimal approximation

In this case there is no need for a decimal representation as the solution is an exact integer.

Key Concepts

Domain of Logarithmic Functions

When solving logarithmic equations, understanding the domain is crucial. Logarithmic functions have a specific domain: the argument (the value inside the logarithm) must be positive. For example, in the equation \(\log_{5}x = 3\), the domain dictates that \(x > 0\). This is because the logarithm of a negative number or zero is undefined.
This positive restriction helps ensure valid solutions only. Any candidate solution that does not satisfy a positive argument must be discarded. Before concluding that a solution is correct, check it falls within this domain.

Exponential Form of Logarithms

To solve logarithmic equations, we often need to transform them into exponential form. This makes it easier to find solutions. The logarithmic equation \(\log_{b}(y) = x\) is equivalent to the exponential equation \(b^x = y\).
Applying this to \(\log_{5}x = 3\), we convert it to \(5^3 = x\). This transformation relies on our understanding that a logarithm tells us the power to which a base must be raised to obtain a number.

• Logarithmic form: \(\log_{b}(y) = x\)
• Exponential form: \(b^x = y\)

Using these forms helps navigate and solve equations more effectively.

Solving Exponential Equations

With the exponential form at hand, solving equations becomes straightforward. In \(5^3 = x\), we need to evaluate the expression on the left hand side. Raising 5 to the power of 3:
\(5 \times 5 \times 5 = 125\). This multiplication of the base by itself three times gives us the value of \(x\).
The process of solving exponential equations like this involves identifying the powers and performing the necessary multiplication. Make sure iteration and steps follow the initial transformation from the log form, closely leading to an exact integer solution in this case.

Exact and Decimal Solutions

Solving equations can result in exact or decimal solutions. In this case, the equation \(5^3 = x\) gives an exact integer solution: 125. Sometimes, the result might not be an integer but an irrational or a long decimal number.
When a calculator is used, particularly with non-integer results, a decimal approximation might be necessary. The original exercise specifies two decimal places for approximations, ensuring precision yet practicality.
However, as noted, this doesn't apply to our problem—a clear integer solution, making approximations unnecessary. Nevertheless, understanding both forms ensures thorough preparation for diverse logarithmic and exponential equation scenarios.