Problem 13 - Precalculus : Building Concepts and Connections

Question

Solve the exponential equation. Round to three decimal places, when needed. $$2^{x}=5$$

Step-by-Step Solution

Answer

So, the solution to the equation is \(x \approx 2.322\).

Step 1: Convert the exponential equation to a logarithmic equation

In this case, the given equation is \(2^{x}=5\). The base of the exponent is 2. In the logarithmic form the base of the log is also 2. Therefore the equation can be written as \(\log_{2}(5) = x\).

Step 2: Find the value of x by evaluating the logarithm

Use a calculator to evaluate \(\log_{2}(5)\). Round the answer to three decimal places, when needed. This is the solution for x.

Key Concepts

Logarithmic Form

Exponential equations, like the one presented in the exercise \(2^{x}=5\), consist of a base raised to a power that gives a certain number. However, when solving these equations, it's often easier to maneuver with an alternative expression. This is where the logarithmic form comes into play.

Think of the logarithm as a detective that answers the question: 'To what power do we need to raise the base to get this number?' In other words, when you see \(2^{x}=5\), the logarithmic form will ask, 'What power of 2 gives us 5?' The notation we use for this is \(\log_{2}(5)\), which means 'the logarithm of 5 with base 2.'

The equality between the exponential and logarithmic forms is a cornerstone in mathematics, allowing us to swap confidently between saying '2 raised to what equals 5?' and 'the logarithm (base 2) of 5 is what?' The answer to both is the elusive value of \(x\) we seek to find.

Exponential to Logarithmic Conversion

Understanding how to shuffle between exponential and logarithmic forms is an essential skill in algebra. This conversion follows a simple rule: the base of the exponent becomes the base of the logarithm, while the exponential result becomes the true argument of the logarithm.

In the case of the equation \(2^{x}=5\),

Converting to Logarithmic Form

we look at the base \(2\) and the outcome \(5\), and we can rewrite it as \(\log_{2}(5) = x\), which effectively gives us a way to isolate \(x\) and bring it down from its position as an exponent, making it easier to handle and solve.

This transformation from exponential to logarithmic form allows us to apply a range of algebraic tools, which are not readily available when dealing with exponential equations directly. It's a bit like finding the right key for a locked door; once we convert it, we have the means to unlock the value of \(x\).

Calculator Use in Logarithms

Once you've converted an exponential equation to its logarithmic counterpart, the next challenge is to calculate the actual value. For most of us, this isn't something we can do in our heads or even on paper. Instead, calculators step into the limelight.

With the equation \(\log_{2}(5) = x\), you'll want to grab a scientific calculator to help you out. Most calculators have a log function, but it's typically configured to use base 10 or the natural logarithm base \(e\).

Finding the Logarithm with a Different Base

For bases like 2 in our example, you may need to use a little workaround known as the change of base formula, which states that \(\log_{b}(a) = \frac{\log(a)}{\log(b)}\), where both logarithms on the right-hand side are to a common base, whether 10 or \(e\).

This formula allows you to input the value into your calculator using the base your calculator recognizes. Once you've pressed the right buttons and followed the correct procedure, it provides you with a numerical value for \(x\), rounded to the required number of decimal places. It's important to remember to check your calculator's instructions since the key sequence may vary between models and brands.