Logarithm Problem: Finding the Value of x

This problem asks us to find the value of 'x' in a given logarithmic equation. Understanding the fundamental definition of logarithms is crucial to solving this type of problem.

Logarithm Definition Explained

A logarithm is the inverse operation to exponentiation. It answers the question: "To what power must we raise a base to get a certain number?"

• The general form of a logarithm is \({\log_b}(a) = c\).
• This expression is equivalent to the exponential form: \({b^c} = a\).
• Here, 'b' is the base, 'a' is the argument (or number), and 'c' is the exponent (or logarithm).

Solving the Logarithm Equation Step-by-Step

We are given the logarithm equation: \({\log _{\rm{x}}}\left( {\frac{5}{7}} \right) = - \frac{1}{3}\).

1. Convert to Exponential Form

Using the definition of logarithm (\({\log_b}(a) = c \implies {b^c} = a\)), we can rewrite the given equation in its exponential form:

• Here, the base \(b = {\rm{x}}\).
• The argument \(a = \frac{5}{7}\).
• The exponent \(c = - \frac{1}{3}\).

So, the exponential form becomes:

\({{\rm{x}}^{ - \frac{1}{3}}} = \frac{5}{7}\)

2. Isolate x by Raising to a Power

To find the value of x, we need to eliminate the exponent \( - \frac{1}{3}\). We can do this by raising both sides of the equation to the power of -3. This is because \({{\left( {{a^m}} \right)}^n} = {a^{m \times n}}\), and \(\left( { - \frac{1}{3}} \right) \times \left( { - 3} \right) = 1\).

\({{\left( {{{\rm{x}}^{ - \frac{1}{3}}}} \right)}^{ - 3}} = {{\left( {\frac{5}{7}} \right)}^{ - 3}}\)

Simplifying the left side:

\({{\rm{x}}^{{\left( { - \frac{1}{3}} \right) \times \left( { - 3} \right)}}} = {{\rm{x}}^1} = {\rm{x}}\)

Now, let's simplify the right side. Recall that \({a^{ - n}} = \frac{1}{{{a^n}}}\) and \({\left( {\frac{a}{b}} \right)^{ - n}} = {\left( {\frac{b}{a}} \right)^n}\).

\({{\left( {\frac{5}{7}} \right)}^{ - 3}} = {{\left( {\frac{7}{5}} \right)}^3}\)

3. Calculate the Final Value of x

Now, we cube the numerator and the denominator:

\({\rm{x}} = \frac{{{7^3}}}{{{5^3}}}\)

Calculate the cubes:

• \({7^3} = 7 \times 7 \times 7 = 49 \times 7 = 343\)
• \({5^3} = 5 \times 5 \times 5 = 25 \times 5 = 125\)

Therefore, the value of x is:

\({\rm{x}} = \frac{{343}}{{125}}\)

Comparing with Options

Let's compare our calculated value of x with the given options:

Our calculated value \({\rm{x}} = \frac{{343}}{{125}}\) matches Option 1.