Problem 3 - A Graphical Approach to Precalculus with Limits

Question

Solve each equation. Express all solutions in exact form. Do not use a calculator. $$2\left(10^{x}\right)=14$$

Step-by-Step Solution

Answer

The solution is \( x = \log(7) \).

Step 1: Isolate the Exponential Term

Start by isolating the term with the exponent, which is \(10^x\). To do this, divide both sides of the equation by 2: \( \frac{2(10^x)}{2} = \frac{14}{2} \). This simplifies to \( 10^x = 7 \).

Step 2: Apply Logarithms to Both Sides

To solve for \(x\), apply the logarithm to both sides of the equation. Use the common logarithm (base 10) since the base of the exponential expression is 10: \( \log(10^x) = \log(7) \).

Step 3: Apply Logarithmic Identity

Use the property of logarithms that states \( \log(a^b) = b \cdot \log(a) \). Therefore, \( x \cdot \log(10) = \log(7) \). Since \( \log(10) = 1 \), this simplifies to \( x = \log(7) \).

Step 4: Express the Solution

The solution to the equation is \( x = \log(7) \), which is the exact form required as per the exercise instructions.

Key Concepts

Understanding Logarithms

Logarithms are mathematical tools used to solve equations involving exponential expressions. They are the inverse operation of exponentiation, much like subtraction is to addition. In simpler terms, if you know the value of a number raised to a power and want to find the exponent, you use a logarithm. Logarithms help you "bring down" the exponent so that you can solve for it directly. The common logarithm we often use has a base of 10, which is why we simply write it as \(\log\) without specifying the base. For instance, when we have an equation with \(10^x\), taking the logarithm of both sides allows us to manage and solve for \(x\) easily. Key properties of logarithms include:

• \(\log(a \cdot b) = \log(a) + \log(b)\)
• \(\log(\frac{a}{b}) = \log(a) - \log(b)\)
• \(\log(a^b) = b \cdot \log(a)\)

These properties are essential in simplifying logarithmic expressions and solving related equations.

Finding Exact Solutions Through Logarithms

When tasked with finding an exact solution, it means we're looking for a precise result in mathematical form rather than a numerical approximation. This often involves leaving expressions in their logarithmic or symbolic form. Taking logarithms, as shown in the solution, aids this process immensely. By applying logarithms, the power of 10 becomes more manageable. The original equation of \(10^x = 7\) becomes \(x \cdot \log(10) = \log(7)\). Given that \(\log(10) = 1\), you can directly solve for \(x\) without other numerical substitutions. Thus, the solution is \(x = \log(7)\), keeping it in exact form as instructed. Expressing solutions this way leans heavily on your understanding of logarithmic identities and requires accurate algebraic manipulation. While exact solutions might seem abstract, they are crucial when precision is key.

Isolating the Variable

Isolating the variable is a fundamental step in solving equations, aiming to get the variable alone on one side of the equation. This process makes it possible to find its value. In our exercise with \(2(10^x) = 14\), isolating involves getting \(10^x\) by itself. This was achieved by dividing both sides of the equation by 2. By doing this, we simplify the equation to \(10^x = 7\). The significance of isolating the variable lies in simplifying the equation to a point where we can apply further mathematical operations, such as logarithms.This step is universally essential in equations involving not only exponents but any form of algebraic expressions. Successfully isolating variables can often dictate the ease or complexity of solving the equation. It's your building block towards solving for the unknown!