Problem 31 - Beginning and Intermediate Algebra

Question

Solve the logarithmic equation algebraically. Then check using a graphing calculator. $$\log _{5} x=4$$

Step-by-Step Solution

Answer

To solve the logarithmic equation \(\log_{5} x = 4\) algebraically, rewrite it in exponential form as \(5^4 = x\). Evaluate the exponent to find \(x = 625\). To check the solution, use a graphing calculator to plot \(y = \log_{5}x\) and a horizontal line at \(y = 4\). The graphs intersect at \((625, 4)\), confirming our algebraic solution \(x = 625\).

Step 1: Rewrite in exponential form

To rewrite the given equation in exponential form, we need to recall that \(\log_{a} b = c\) is equivalent to \(a^c = b\). So, the equation \(\log_{5} x = 4\) can be rewritten as \(5^4 = x\).

Step 2: Evaluate the exponent

Now that we have the equation in exponential form, we can evaluate the exponent. We have: \[5^4 = 5 \times 5 \times 5 \times 5 = 625\]

Step 3: Find the value of x

Now we have evaluated the exponent, and we get the equation: \[x = 625\] So the value of x is 625.

Step 4: Check with a graphing calculator

Using a graphing calculator, plot the logarithmic function \(y = \log_{5}x\) and a horizontal line at \(y = 4\). You should notice that these two graphs intersect at the point \((625, 4)\), which confirms our algebraic solution that \(x = 625\).

Key Concepts

Exponential Form

When dealing with logarithmic equations, one helpful method is converting them into exponential form. The relationship between logarithms and exponents can be remembered with the formula: if \( \log_{a} b = c \), this is the same as saying \( a^c = b \). This conversion unveils the base and exponent, making the problem simpler to solve. For instance, in the equation \( \log_{5} x = 4 \), the base is 5, the logarithm results in the exponent 4, and this equals \( x \). Therefore, in exponential form, this equation becomes \( 5^4 = x \). This conversion is a pivotal step in solving logarithmic equations as it transforms an expression that may seem complex at first into a more relatable, solvable form. By rewriting the equation, you can then evaluate the exponent directly. In our example, calculating \( 5^4 \) means multiplying 5 by itself four times, resulting in \( 625 \). Thus, \( x = 625 \). Understanding this transformation makes it easier for you to tackle logarithmic equations consistently by applying the power of exponents.

Logarithmic Functions

Logarithmic functions are essential in mathematics, especially when dealing with growth and decay processes, like compound interest or population dynamics. A logarithmic function with base \( a \) is typically expressed as \( f(x) = \log_{a} x \). These functions are inverses of exponential functions, meaning their outputs determine the power to which the base must be raised to obtain the input.For the equation \( \log_{5} x = 4 \), the function involved is \( y = \log_{5} x \). Here, the function is used to "undo" the operation of raising 5 to a power, which confirms the logarithmic identity where the output of the function (\( y = 4 \)) reflects the exponent. Students often encounter log functions in the context of solving exponential equations and vice-versa. Knowing how to transition between logarithmic and exponential forms (as discussed earlier) is crucial. It enhances your understanding of the relationship between the two forms, fortifying skills required to solve broader sets of mathematical problems. Understanding logarithmic functions also involves getting familiar with their graphical representation, which aids in visualizing how inputs relate logarithmically to outputs on a coordinate plane.

Graphing Calculator

A graphing calculator is a versatile tool that assists in verifying mathematical solutions by visualizing equations graphically. When you've solved a logarithmic equation algebraically, plotting the function on a graphing calculator can provide confirmation. This dual approach of algebraic and graphical verification consolidates your understanding and boosts your confidence in your solution.To verify the solution for \( \log_{5} x = 4 \) using a graphing calculator, graph the function \( y = \log_{5} x \) alongside a horizontal line represented by \( y = 4 \). The intersection point of the graphs should correspond to the algebraic solution. In our case, these graphs intersect at \( (625, 4) \), confirming that \( x = 625 \) is correct. Graphing calculators not only support solving equations but also help in visualizing functions, which is critical for solving more complex problems and improving comprehension of logarithmic behavior. They effectively demonstrate how equations translate to graphs, making them an invaluable educational aid.