Question

If \(\log _{5} x=2,\) then determine \(\log _{5} 125 x\).

Short answer

\text{log}_{5} 125 x = 5.

Step-by-step solution

- Understand given data

We are given that \(\text{log}_{5} x = 2\), which we will use to find \(\text{log}_{5} 125x\).

- Use definition of logarithms

According to the definition, if \(\text{log}_{5} x = 2\), then it means \(x = 5^2 = 25\).

- Expand the expression

To find \(\text{log}_{5} 125x\), we first recognize that \(125 = 5^3\). Hence, we need to calculate \(\text{log}_{5}(5^3 \times x)\).

- Use logarithm properties

Using the property \(\text{log}_a(bc) = \text{log}_a b + \text{log}_a c\), we can write: \[ \text{log}_{5}(5^3 \times x) = \text{log}_{5}(5^3) + \text{log}_{5}(x). \]

- Simplify

We know that \( \text{log}_{5}(5^3) = 3 \) and from the given data, \( \text{log}_{5}(x) = 2\). Thus, \[ \text{log}_{5} (125x) = 3 + 2 = 5. \]

Key concepts

Logarithm Properties

Logarithms have several properties that make them useful in solving mathematical problems.
One key property is the **product rule**: \[\log_a(bc) = \log_a b + \log_a c\]\
This means that the logarithm of a product is the sum of the logarithms of the factors.
Another property is the **power rule**: \[\log_a(b^k) = k \cdot \log_a b\]\
This shows that a logarithm of a number raised to a power can be simplified by multiplying the power by the logarithm of the base number.
Additionally, there's the **change of base formula**: \[\log_b a = \frac{\log_c a}{\log_c b}\]\
This allows converting between different bases, useful when calculators or certain mathematical contexts require a specific base. These properties are critical for simplifying logarithmic expressions and solving logarithmic equations.

Definition of Logarithms

A logarithm answers the question: 'To what power must a base be raised to get a certain number?' For instance,
\[\log_5 25 = 2\]\
This means that 5 must be raised to the power of 2 to result in 25. It's like asking, '5 to the power of what equals 25?' Logarithms convert exponential expressions into a more manageable form, useful for solving various mathematical problems, especially exponent-related equations.
In mathematical notation, \(\log_a b = c\) if and only if \(a^c = b\). For example,
// Here, \(a\) is the base, \(b\) is the result and \(c\) is the exponent.
This relationship is fundamental to understanding and working with logarithms.

Logarithm Simplification

Simplifying logarithms often involves applying logarithm properties appropriately.
Consider the exercise: \(\log _{5} 125 x\). First, break it down using the product rule:
\