91Ó°ÊÓ

When dealing with logarithmic equations, it can be helpful to convert them into exponential form. This is because many students find exponential expressions easier to manipulate.
A logarithmic equation is generally written as \( \log_b a = c \). This tells us that the exponent \( c \) is what we need to raise the base \( b \) to in order to get \( a \).
In exponential form, it becomes \( b^c = a \). For instance, if we have \( \log(4x - 18) = 1 \), rewriting it in exponential form means using a base of 10. So, the equation changes to \( 10^1 = 4x - 18 \). This simplifies our work, converting the problem into a simple linear equation: \( 10 = 4x - 18 \).
Converting logarithms to exponential form is often the first and most crucial step in solving these types of problems.

Once we have our equation in a simpler linear form, we can easily solve it using basic algebra.
In our example, the converted exponential form equation is \( 10 = 4x - 18 \). To solve for \( x \), we need to isolate it on one side of the equation.

• First, add 18 to both sides: \( 10 + 18 = 4x \). This simplifies to \( 28 = 4x \).
• Next, divide both sides by 4: \( \frac{28}{4} = x \). This simplifies to \( x = 7 \).

With these two steps, we've solved \( x \). Solving linear equations usually only requires a couple of arithmetic operations like addition, subtraction, multiplication, or division.

After finding a solution, it is essential to check that it satisfies the original equation. This step ensures the solution is correct and rules out any calculation errors.
In our example, we substituted \( x = 7 \) back into the original logarithmic equation \( \log(4x - 18) = 1 \).

• Calculate \( 4 \times 7 - 18 \) to see if it equals 10.
• This gives \( 28 - 18 = 10 \). Thereby, \( \log(10) \) should be 1, confirming our solution as correct.

Checking solutions in logarithmic equations is straightforward, often leading to a simple truthful statement, like in this example.