91Ó°ÊÓ

Logarithms are a way to express exponents. They answer the question: 'To what power must a certain base be raised, to obtain a given number?'. For example, if we have \ \ \(\text{log}_{b}(a) = c\), this means that \(b^c = a\). In our exercise, \( \log _{5}(4x-2) = \log _{5} 10 \), the base is 5. We need to find what value of \( x \) makes the equation true. \ \ Remember: Logarithms have three main parts: the base, the exponent (or logarithm), and the argument. Here, the base is 5, the argument is \(4x - 2\) or 10, and the exponent is the logarithm value itself.

One important property of logarithms is that if two logarithms with the same base are set equal, their arguments must also be equal. This is the key property used in our solution. Mathematically, this property is written as: \( \text{log}_b(M) = \text{log}_b(N) \Rightarrow M = N\). \ In our exercise, \( \log _{5}(4 x-2) = \log _{5} 10\), since the base of the logarithms on both sides of the equation is the same (5), we can conclude: \( 4x - 2 = 10 \). From here, it just becomes a matter of solving the equation. \ Another property of logarithms to remember is the change of base formula, but we don’t need it here because the bases in our equation are already the same. For your reference, the change of base formula is: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \).

Solving the equation \( 4x - 2 = 10\) involves basic algebra steps. Let’s break this down further. \

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• Step 1: Add 2 to both sides to isolate the term with \(x\): \(4x - 2 + 2 = 10 + 2\) which simplifies to \(4x = 12\).
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• Step 2: Divide both sides by 4 to solve for \(x\): \( \frac{4x}{4} = \frac{12}{4}\) which gives us \( x = 3 \).
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\ Therefore, the solution to the original logarithmic equation is \( x = 3 \). Simple steps like these form the building blocks of advanced math, so practice solving equations step-by-step to build a solid foundation.