Question

Solve each logarithmic equation. Express irrational solutions in exact form. $$ \log _{4}(x+4)=\log _{4} 15 $$

Short answer

x = 11

Step-by-step solution

- Understand the Properties of Logarithms

Determine if the logarithmic equation can be simplified using properties of logarithms. In this case, we have \(\text{log}_{4}(x+4) = \text{log}_{4} 15\). Since the bases of the logarithms on both sides of the equation are the same, we can set the arguments equal to each other.

- Set the Arguments Equal

Using the property that if \(\text{log}_{b}(A) = \text{log}_{b}(B)\), then \A = B\, we set the arguments of the logarithms equal to each other. Thus, \(x + 4 = 15\).

- Solve for x

Solve the equation from Step 2 for \(x\). Subtract 4 from both sides to isolate \(x\) on one side. \(x + 4 - 4 = 15 - 4\), which simplifies to \(x = 11\).

- Verify the Solution

Substitute \(x = 11\) back into the original equation to verify the solution. The original equation is \(\text{log}_{4}(x + 4) = \text{log}_{4} 15\). Substituting \(x = 11\) gives \(\text{log}_{4}(11 + 4) = \text{log}_{4} 15\), and both sides of the equation simplify to \(\text{log}_{4} 15\), thus verifying the solution.

Key concepts

Logarithm Properties

Logarithms have unique properties that make solving equations easier. One important property is: if the logarithmic functions on both sides of an equation have the same base, then their arguments must be equal.

For example, in the problem \(\text{log}_{4}(x+4) = \text{log}_{4} 15\), since the base is 4 on both sides, we can simplify by equating the arguments: \x + 4 = 15\.

This property significantly reduces the complexity of solving logarithmic equations. Additionally, understanding how to manipulate and combine logarithmic expressions is key to mastering this topic. Stay mindful of the base of the logarithms when applying these properties.

Equation Solving

Once we leverage the properties of logarithms, solving the equation becomes straightforward.

In the above example, after equating the arguments \(x + 4 = 15\), the next step is to isolate \(x\). We do this by performing simple algebraic operations: subtract 4 from both sides: \(x + 4 - 4 = 15 - 4\), simplifying to \(x = 11\).

This step emphasizes the importance of basic algebra skills, as solving for the variable often requires straightforward calculations. Always ensure each step maintains the balance of the equation, as this keeps the solution valid.

Verification of Solutions

Verifying your solution is crucial to confirming its accuracy. Substitute the found value back into the original equation to check if it satisfies the equation.

For \(x = 11\), place it back into \(\text{log}_{4}(x + 4) = \text{log}_{4} 15\):
\(\text{log}_{4}(11 + 4) = \text{log}_{4} 15\).

Simplifying the left side: \(11 + 4 = 15\), so the equation reads \(\text{log}_{4} 15 = \text{log}_{4} 15\), which confirms that our solution is correct.

Never skip the verification step, as it ensures your solution is not only mathematically correct but also satisfies the given equation. This practice builds confidence and accuracy in problem-solving.