Question

Solve each equation. $$ \log _{7}\left(x^{2}+4\right)=2 $$

Short answer

x = \pm 3\sqrt{5}

Step-by-step solution

Rewrite in Exponential Form

Convert the logarithmic equation \(\text{log}_{7}(x^{2} + 4) = 2\) into its exponential form using the definition of a logarithm: \[ y = \text{log}_{b}(x) \rightarrow b^{y} = x \] for \[ \text{log}_{7}(x^{2} + 4) = 2 \rightarrow 7^{2} = x^{2} + 4 \]

Simplify the Exponential Equation

Calculate the exponent value: \[ 7^{2} = 49 \] Now the equation becomes: \[ 49 = x^{2} + 4 \]

Isolate the Variable Term

Subtract 4 from both sides of the equation to isolate \( x^{2} \): \[ 49 - 4 = x^{2} \rightarrow 45 = x^{2} \]

Solve for x

Take the square root of both sides of the equation to solve for \( x \): \[ x = \pm \sqrt{45} \] Simplify the square root: \[ x = \pm 3\sqrt{5} \]

Key concepts

Exponential Form

When we encounter a logarithmic equation, converting it to its exponential form is key. Think of a logarithm as a way to find what power you need to raise a certain base to get a number. For example, in the equation \[\log_{7}(x^{2} + 4) = 2\], we can use the definition of a logarithm, which tells us that \[y = \log_{b}(x) \rightarrow b^{y} = x\]. This means our equation translates to \[7^{2} = x^{2} + 4\]. By converting the log equation to exponential form, it often becomes easier to handle and solve. This step is fundamental because it sets the groundwork for isolating variables and proceeding with algebraic manipulation.

Solving Equations

Once we have our equation in exponential form, the next step is to simplify it. In our example, \[7^{2} = 49\], so the equation becomes \[49 = x^{2} + 4\]. Solving equations typically involves isolating the variable you’re solving for. Here, we need to isolate \[x^{2}\] by subtracting 4 from both sides, giving us \[45 = x^{2}\]. Remember, each step you take to simplify an equation moves you closer to finding the variable's value. These steps are essential as they build up to the final solution through straightforward arithmetic operations.

Square Roots

The next step is to solve for \[x\] by taking the square root of both sides of the equation. Taking the square root is necessary when dealing with equations involving \[x^{2}\]. For our specific case, we get \[x = \pm \sqrt{45}\]. Simplifying \[\sqrt{45}\] further, we notice that 45 can be expressed as 9 * 5; thus, \[\sqrt{45} = \sqrt{9 \cdot 5} = 3\sqrt{5}\]. So, \[x = \pm 3\sqrt{5}\]. Always remember that when taking square roots, you must consider both the positive and negative solutions. This is because both \[ (3\sqrt{5})^{2}\] and \[(-3\sqrt{5})^{2}\] will give you 45.

Logarithmic Functions

Understanding logarithmic functions is crucial for solving logarithmic equations. A logarithm essentially undoes what an exponent does. For example, if you know that \[7^{2} = 49\], you can find the logarithm by asking which power you have to raise 7 to get 49, which is 2, thus \[ \log_{7}(49) = 2\]. Logarithmic functions have distinct properties that allow us to solve for unknowns effectively. Knowing how to convert between exponential and logarithmic forms helps in tackling more complex equations and fosters a deeper understanding of their applications in various fields such as science, engineering, and finance.