Problem 498 - Precalculus

Question

Rewrite \(\log _{17}(4913)=x\) as an equivalent exponential equation.

Step-by-Step Solution

Answer

The equivalent exponential equation is \(17^x = 4913\).

Step 1: Identify the Components of the Logarithmic Equation

The original equation is given as \(\log_{17}(4913) = x\). Here, 17 is the base of the logarithm, 4913 is the argument, and \(x\) is the exponent in the equivalent exponential form.

Step 2: Rewrite in Exponential Form

To convert the logarithmic equation \(\log_{17}(4913) = x\) into an exponential equation, remember that this represents the statement: 17 raised to the power \(x\) equals 4913. Therefore, rewrite it as \(17^x = 4913\).

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear in the exponents, rather than as base numbers. This makes them particularly powerful for calculating rapid growth or decay, such as in finance or population models. To solve these equations, we often need to find what power a given base number must be raised to reach a specific value.
For example, in the equation \(17^x = 4913\), our task is to determine the value of \(x\), the exponent. By rewriting logarithmic equations as exponential equations, as seen in the given problem, we gain a new perspective on reaching our solution. Here, you're effectively translating the logarithmic problem into one where you raise 17 to the correct power to achieve 4913. This is a fundamental step in understanding how exponential changes work in various scientific and practical applications.

Logarithmic Equations

Logarithmic equations inversely relate to exponential equations. Where exponential equations involve raising a number to a power, logarithmic equations try to find that power given a base and its associated result. Logarithms are particularly useful in identifying the rate of growth in models where numbers get too large, too fast.
Consider the logarithmic equation from the problem, \(\log_{17}(4913) = x\). This reads as "the power to which 17 must be raised to equal 4913 is \(x\)." Solving logarithmic equations often involves rearranging or converting them into exponential form. This conversion helps to simplify complex relationships, making logarithmic equations a vital tool in fields such as engineering and computer science.

Base of a Logarithm

The base of a logarithm is a critical component in both logarithmic and exponential equations. It indicates the number that is repeatedly multiplied in an exponential function. Understanding the base is key to unlocking the solution to any logarithmic or exponential problem.
In the example equation \(\log_{17}(4913) = x\), the base is 17. This tells us that all calculations involve powers of 17, which is central to solving the equation. Choosing the correct base is important as it influences the outcome of the equation. Different bases can result in different values of \(x\). By consistently practicing with equations of varying bases, you become adept at recognizing how different bases function and solving problems efficiently.