91Ó°ÊÓ

Logarithms are mathematical operations that help us with simplifying complex calculations. One of the fundamental properties of logarithms is if two logs with the same base are equal, the expressions inside them (called the arguments) must also be equal. This is because the logarithmic function is one-to-one. In our original exercise, both sides of the equation have logs with the same base, 4. This allows us to set their arguments equal. In simpler terms, the property tells us:

• If \(\log_b(A)=\log_b(B)\), then \(A=B\) when both are defined.

This property is crucial when solving logarithmic equations as it lets us remove the logs and makes the equation more straightforward to solve.
Applying it to the solved exercise, we could directly equate the arguments: \(5y + 2 = 10\). From here, it's a matter of algebraic manipulation to find the solution.

When dealing with logarithms in equations, recognizing their equality can simplify the process significantly. If two logarithms are equal, under the same base, as seen in \(\log_4(5y+2)=\log_4(10)\), we can immediately focus on their arguments.

• This means our goal is to check if the numbers inside the logs are equal.
• Remember, they must be defined within the logarithm’s domain, meaning they must be positive if we are working with real numbers.

Once the logs are removed (by using the equality property), you are only left with figuring out the correct approach to match the arguments. In this specific exercise, the path forward is clear, and it becomes a basic algebra problem.

Algebraic manipulation involves using operations like addition, subtraction, multiplication, and division to rearrange equations and solve for unknowns. In solving the given logarithmic equation, after setting \(5y+2=10\), our task was to isolate the variable, \(y\).

• Subtract 2 from both sides to counteract the +2 in the equation: \(5y=10-2\).
• This simplifies to \(5y=8\).
• Finally, to solve for \(y\), divide both sides by 5: \(y=\frac{8}{5}\).

Each step requires careful attention to ensure every action maintains equality. Rechecking each step helps in verifying the correctness of your operations. This process shows how effective algebraic strategies can simplify and solve what might initially appear as complex equations.