Problem 4 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Express all solutions in exact form. Do not use a calculator. $$5\left(10^{3 x}\right)-4=6$$

Step-by-Step Solution

Answer

\(x = \frac{\log_{10}(2)}{3}\)

Step 1: Isolate the Exponential Term

Start by moving the constant term on the left side of the equation to the right side. So, add 4 to both sides to get:\[5(10^{3x}) = 10\]

Step 2: Divide by the Coefficient

Now, divide both sides of the equation by 5 to isolate the exponential term:\[10^{3x} = 2\]

Step 3: Apply the Logarithm

To solve for \(x\), take the logarithm of both sides. Using common logarithm (base 10), the equation becomes:\[\log_{10}(10^{3x}) = \log_{10}(2)\]

Step 4: Use the Power Rule of Logarithms

Apply the power rule of logarithms, which states that \(\log_{b}(a^n) = n \cdot \log_{b}(a)\). Therefore:\[3x \cdot \log_{10}(10) = \log_{10}(2)\]

Step 5: Simplify the Logarithm

Since \(\log_{10}(10) = 1\), the equation simplifies to:\[3x = \log_{10}(2)\]

Step 6: Solve for x

Finally, divide both sides by 3 to solve for \(x\):\[x = \frac{\log_{10}(2)}{3}\]

Key Concepts

Isolating the Exponential Term

When tackling exponential equations, the goal is often to get the tricky part—the exponential term—on its own. This involves isolating the term to make the equation easier to handle. If we have an equation like \(5(10^{3x}) - 4 = 6\), our first step is to shift other constants away from the exponential part.

• We start by adding or subtracting terms to both sides of the equation so that the exponential component is by itself.
• In this example, adding 4 to both sides moves the constant term on the left side, resulting in \(5(10^{3x}) = 10\).

The next step is to clear any coefficients that might be multiplying the exponential term. This is done using division:

• Divide each side of the equation by the coefficient to leave the exponent in a clean state.
• In our example, dividing both sides by 5 gives us \(10^{3x} = 2\).

With the exponential term isolated, you can proceed with solving the equation using logarithms.

Using Logarithms

Logarithms serve as powerful tools for dealing with exponential terms, thanks to their unique properties that simplify exponential expressions. When an exponential term is alone, as \(10^{3x} = 2\), logarithms can be applied to both sides of the equation:

• Applying logarithms helps us "bring down" the exponent to a more operable level, converting it into a coefficient.
• For base 10 exponentials, we use the common logarithm (logarithm with base 10), so we begin with \(\log_{10}(10^{3x}) = \log_{10}(2)\).

Using logarithms effectively allows us to transform the equation into a linear form, where we can apply familiar algebraic techniques to find a solution.

Power Rule of Logarithms

The power rule of logarithms is an essential tool that helps simplify expressions with exponents inside logarithms. It states that \(\log_{b}(a^n) = n \cdot \log_{b}(a)\). This rule lets us pop the exponent down as a multiplier, simplifying equations significantly.

• In the equation \(\log_{10}(10^{3x}) = \log_{10}(2)\), we apply the power rule: \(3x \cdot \log_{10}(10) = \log_{10}(2)\).
• The beauty of this rule lies in its ability to convert the exponential term to a simple multiplication, making future steps a breeze.

We often find that certain logarithms simplify further. In this scenario, \(\log_{10}(10)\) simplifies to 1, as any number to the power of 1 is itself. This gives us:

• \(3x = \log_{10}(2)\), which finally transforms into \(x = \frac{\log_{10}(2)}{3}\) when we solve for \(x\).

Using the power rule allows us to break down and simplify problems, providing an exact solution without the need for approximations.