Question

Solve for \(x\) a) \(x^{\frac{2}{\log x}}=x\) b) \(\log x^{\log x}=4\) c) (log \(x)^{2}=\log x^{2}\)

Short answer

a) No solution. b) x = 100 or x = 0.01. c) x = 1 or x = 100.

Step-by-step solution

Solve Part (a)

Consider the equation: \[ x^{\frac{2}{\log x}} = x \] Take the natural logarithm of both sides: \[ \ln\left( x^{\frac{2}{\log x}} \right) = \ln(x) \] Use the power rule of logarithms \[ \frac{2}{\log x} \cdot \ln(x) = \ln(x) \] Simplify by isolating \( \ln(x) \): \[ \frac{2 \ln(x)}{\ln(x)} = 1 \] This simplifies to: \[ 2 = 1 \] This is a contradiction, so there is no solution for part (a).

Solve Part (b)

Consider the equation: \[ \log(x^{\log x}) = 4 \] Use the power rule of logarithms: \[ \log x \cdot \log x = 4 \] Simplify to: \[ (\log x)^2 = 4 \] Take the square root of both sides: \[ \log x = \pm2 \] Therefore, solve for \(x\): \[ x = 10^2 \quad \text{or} \quad x = 10^{-2} \] Which gives: \[ x = 100 \quad \text{or} \quad x = 0.01 \]

Solve Part (c)

Consider the equation: \[ (\log x)^{2} = \log x^{2} \] Use the power rule of logarithms: \[ (\log x)^{2} = 2 \log x \] Let \( y = \log x \). The equation becomes: \[ y^2 = 2y \] Solve the quadratic equation: \[ y^2 - 2y = 0 \] Factor out \( y \): \[ y(y-2) = 0 \] Therefore, solve for \( y \): \[ y = 0 \quad \text{or} \quad y = 2 \] Recall \( y = \log x \), so: \[ \log x = 0 \quad \text{or} \quad \log x = 2 \] Solving for \( x \): \[ x = 10^0 = 1 \quad \text{or} \quad x = 10^2 = 100 \]

Key concepts

solving logarithmic equations

Logarithmic equations are equations that involve a logarithm with a variable. Solving them often requires using properties of logarithms to simplify and isolate the variable.

Here are key steps for solving logarithmic equations:

• Use properties of logarithms to combine or break apart log expressions.
• Take the logarithm of both sides if it helps to simplify the equation.

Consider part (a) of the exercise, which demonstrates that some equations might not have a solution. After using the power rule and simplifying, we find no solution since we end up with a contradiction.

natural logarithm

The natural logarithm, denoted as \(\text{ln}\), is the logarithm to the base \(e\), where \(e\) is approximately 2.718. Natural logarithms are commonly used in higher mathematics.

In the exercise, we used the natural logarithm to help solve part (a). Taking the natural log of both sides of the equation helps to simplify the problem since the \(\text{ln}\) function is the inverse of the exponential function.

For example, in solving \(x^{\frac{2}{\text{log } x}}=x\), taking the natural logarithm of both sides makes it easier to use the power rule of logarithms to further simplify the equation.

quadratic equations

Quadratic equations are polynomial equations of degree 2, which means they have the form \(ax^2 + bx + c = 0\).

Solving quadratic equations can involve factoring, using the quadratic formula, or completing the square.

In part (c) of our exercise, we converted the logarithmic equation into a quadratic one: \(y^2 = 2y\).

Rewriting it as \(y^2 - 2y = 0\), we could factor it as \(y(y-2) = 0\), which led us to solutions \(y = 0\) or \(y = 2\). Then, substituting back \(y = \text{log } x\), we found the solutions for \(x\).

power rule of logarithms

The power rule of logarithms states that \(\text{log}_b(a^c) = c \text{ log}_b(a)\).

This rule is particularly useful for simplifying logarithmic expressions where the argument is raised to a power.

For instance, in part (b) of the exercise, we utilized the power rule on \(\text{log}(x^{\text{log} x}) = 4\). Applying the power rule allowed us to rewrite the equation as \( \text{log} x \times \text{log} x = 4 \), making it straightforward to solve for \(x \) by taking the square root on both sides.

Understanding and applying the power rule is essential in many logarithmic problem-solving contexts. It simplifies the process and helps unveil the algebraic nature of the logarithmic expressions.