Problem 98 - Algebra and Trigonometry

Question

Solve each equation. $$ 5^{x^{2}}=50 $$

Step-by-Step Solution

Answer

The solutions for \(x\) are \( \pm \sqrt{\frac{\ln(50)}{\ln(5)}}\)

Step 1: Write the given equation

The given equation is \(5^{x^{2}} = 50\)

Step 2: Apply natural logarithm on both sides

Applying the natural logarithm (ln) function to both sides to simplify the equation, the equation becomes \(\ln (5^{x^{2}}) = \ln(50)\)

Step 3: Use the power rule of logarithms

By the power rule of logarithms, an exponent can be brought to the front of the logarithm as a coefficient, so the equation can be rewritten as \(x^{2}\ln(5) = \ln(50)\)

Step 4: Isolate the \(x^{2}\) term

Dividing both sides of the equation by \(\ln(5)\) to isolate the \(x^{2}\) term, we get \(x^{2} = \frac{\ln(50)}{\ln(5)}\)

Step 5: Solve for \(x\)

Finally, to solve for \(x\), taking square roots of both sides of the equation will isolate \(x\), resulting in \(x = \pm \sqrt{\frac{\ln(50)}{\ln(5)}}\)

Key Concepts

Natural Logarithm

The natural logarithm, denoted as \(\ln\), is a special logarithm with the base \(e\), where \(e\) is approximately 2.71828. It's often used when phenomena involve growth and decay processes, like population growth or radioactive decay.

When solving exponential equations like \(5^{x^2} = 50\), using the natural logarithm simplifies expressions involving powers. This is because the natural logarithm can help "bring down" exponents, turning complex powers into manageable coefficients. With exponentials, applying \(\ln\) to both sides of an equation allows us to handle the powers easily. For example:

• \(\ln (5^{x^2}) = x^2 \ln(5)\)

This simplification occurs because of the relationship expressed through the power rule of logarithms, essentially making it simpler to isolate variables.

Power Rule of Logarithms

The power rule of logarithms states that the logarithm of a number raised to an exponent can be expressed as the product of the exponent and the logarithm of the number. In other words, \(\ln(a^b) = b \cdot \ln(a)\). This rule is incredibly useful when dealing with equations where the unknown variable is in an exponent.

For instance, in our example, starting with \(\ln (5^{x^2}) = \ln(50)\), using the power rule, we rewrite this as \(x^2 \cdot \ln(5) = \ln(50)\).

• This manipulation simplifies solving for \(x^2\), since \(x^2\) is no longer trapped in the exponent.

Applying the power rule of logarithms regularly in problem-solving helps break down complex exponential equations into simpler algebraic forms.

Solving Quadratic Equations

Once the equation has been simplified using logarithms, you're often left with a quadratic equation. A common form might look like \(x^2 = \frac{\ln(50)}{\ln(5)}\). Solving quadratic equations involves finding the values of \(x\) that satisfy the equation.

Here, to solve for \(x\), you would:

• Take the square root of both sides to solve for \(x\), yielding \(x = \pm \sqrt{\frac{\ln(50)}{\ln(5)}}\).

Quadratics can often have two solutions, as seen with the positive and negative square root. In this scenario, the quadratic nature leads to recognizing that solving quadratics often requires considering multiple potential solutions.