Question

Solve the equation. \(\log _6(5 x+9)=\log _6 6 x\)

Short answer

The value of \(x\) that satisfies the given logarithmic equation is \(9\).

Step-by-step solution

Understand the equation and apply logarithm property

The given equation is \(\log _6(5 x+9)=\log _6 6 x\). Here, both the logarithms have the same base, which is 6. So, applying the property of logarithms, if \(\log_b a = \log_b c\), then \(a = c\). Therefore, the equation simplifies to \(5x + 9 = 6x\).

Solve the Equation

To find the value of \(x\), subtract \(5x\) from both sides of the equation. This gives: \(6x - 5x = 9\), simplifying further gives: \(x = 9\).

Key concepts

Logarithm Properties

Understanding logarithm properties is crucial when solving logarithmic equations, as it helps simplify the problem into more manageable pieces. One of the fundamental properties of logarithms is that if two logs have the same base and are equal, then their arguments must also be equal. This is expressed mathematically as \( \log_b(a) = \log_b(c) \implies a = c \). Using this property allows us to equate the arguments of the logarithms directly, bypassing the need to deal with the log function itself.

Other important properties include the product rule \( \log_b(mn) = \log_b(m) + \log_b(n) \), the quotient rule \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \), and the power rule \( \log_b(m^n) = n * \log_b(m) \). Being familiar with these properties can transform complicated-looking equations into simpler algebraic expressions that are much easier to solve.

Algebraic Techniques

When solving logarithmic equations, it's important to employ algebraic techniques effectively. Once you have used the properties of logarithms to simplify the equation, the next step is to isolate the variable you are solving for. This often involves basic algebraic operations such as addition, subtraction, multiplication, division, and sometimes factoring.

In the case of our exercise, we simplified the equation using the logarithm property to obtain \(5x + 9 = 6x\). The next step involves algebraic manipulation. By subtracting \(5x\) from both sides of the equation, we're left with \(x = 9\), effectively isolating the variable and finding the solution. The ability to manipulate these equations algebraically is just as important as understanding logarithms themselves.

Equation Simplification

Equation simplification is a process that involves reducing complexity to make an equation easier to solve. The goal is to transform an equation into its simplest form without changing the solution. This often means getting rid of extraneous information, combining like terms, and simplifying fractions or radicals if they are present.

For logarithmic equations, this simplification process starts with applying logarithmic properties and is followed by algebraic techniques. As seen in our exercise, once we used the property that allowed us to set the arguments equal to each other, simplification was straightforward. The simpler equation \(5x + 9 = 6x\) was then easy to solve without the logarithmic notation. Remember, keeping equations simple is key to finding the right solution without unnecessary confusion or complexity.