Problem 7 - Elementary and Intermediate Algebra

Question

Fill in the blanks. To solve \(5^{x}=2,\) we can take the______ of both sides of the equation to get \(\log 5^{x}=\log 2\).

Step-by-Step Solution

Answer

Take the logarithm.

Step 1: Identify the Operation to Use

To solve the equation \(5^{x}=2\), we need to take the logarithm of both sides. This is because logarithms are helpful in transforming exponential equations into forms where \(x\) can be more easily isolated.

Step 2: Take the Logarithm of Both Sides

We apply the logarithm to both sides of the equation. This gives us \(\log(5^{x}) = \log(2)\). The purpose of this step is to use the logarithmic power rule in the next step.

Step 3: Apply the Logarithmic Power Rule

Use the logarithmic identity \(\log(a^{b}) = b \cdot \log(a)\) to simplify \(\log(5^{x})\) to \(x \cdot \log(5)\). This changes our equation to \(x \cdot \log(5) = \log(2)\).

Step 4: Solve for x

Isolate \(x\) by dividing both sides of the equation by \(\log(5)\). This results in \(x = \frac{\log(2)}{\log(5)}\).

Key Concepts

Understanding Exponential Equations

Exponential equations are types of equations where the unknown variable appears as an exponent. For example, in the equation \(5^x = 2\), \(x\) is the exponent. Solving these equations often requires special techniques because the variable is not in a straightforward position like addition or multiplication. This is where logarithms come into play.

In our specific equation, \(5^x = 2\), the goal is to find the value of \(x\) that makes this equation true. Directly solving for \(x\) is challenging because it is not easy to look at the equation and immediately see what power of 5 equals 2. Therefore, we use logarithms to transform this equation.

In general, when you encounter equations in the form of \(a^x = b\), using logarithms helps us "bring down" the exponent in a way that it can be solved like a regular algebraic equation. This is one of the major reasons logarithms are powerful tools for solving exponential equations.

Applying Logarithmic Identities in Equations

Logarithms have several key identities that help in manipulating and solving equations. One of the primary identities is the power rule: \(\log(a^b) = b \cdot \log(a)\). This is extremely useful when dealing with exponential equations.

In the equation \(\log(5^x)\), by using the power rule, you can rewrite it as \(x \cdot \log(5)\). This shows how the exponent "comes down" from its elevated position. Instead of having \(x\) caught in the exponential form, it turns the equation into a multiplication problem which is much easier to solve.

This identity is rooted in the basic properties of logarithms and exponents, and using it effectively allows us to simplify complex exponential equations into solvable forms. Remembering and applying this identity is crucial when working with logarithms.

Steps for Solving Equations with Logarithms

To solve equations that involve exponents using logarithms, follow a systematic approach:

• **Recognize the Equation Type**: Start by identifying that you have an exponential equation, such as \(a^x = b\).
• **Apply Logarithms**: Use logarithms on both sides of the equation. This effectively transforms the equation from an exponential form to a linear form with respect to the variable \(x\).
• **Use Logarithmic Identities**: Utilize identities like the power rule, \(\log(a^b) = b \cdot \log(a)\), to simplify the equation.
• **Isolate the Variable**: Solve for \(x\) by isolating it on one side of the equation. This often involves dividing by the logarithm of the base (in our case, \(\log(5)\)).

In our example, after taking the log of both sides and applying the identity, the solution is completed by dividing through by \(\log(5)\) to isolate \(x\). The result is \(x = \frac{\log(2)}{\log(5)}\). This systematic approach ensures clarity and accuracy in solving exponential equations.