Problem 19 - Algebra for College Students

Question

Solve each of the equations. $$ 10^{x}=0.1 \quad\\{-1\\} $$

Step-by-Step Solution

Answer

The solution is \( x = -1 \).

Step 1: Understand the equation

The given equation is \( 10^x = 0.1 \). We want to find the value of \( x \) that makes this equation true. Recall that \( 0.1 \) is a decimal equivalent to \( 10^{-1} \), which might suggest a logarithmic property.

Step 2: Convert the equation using logarithms

Recognize that \( 0.1 \) is \( 10^{-1} \). This simplifies the equation to \( 10^x = 10^{-1} \). When the bases on both sides of the equation are the same, you can equate the exponents.

Step 3: Equate the exponents

Since the base is 10 on both sides: \( x = -1 \). Therefore, the value of \( x \) is \( -1 \).

Step 4: Verify the solution

Substitute \( x = -1 \) back into the original equation: \( 10^{-1} \) should equal \( 0.1 \). Indeed, \( 10^{-1} = 0.1 \), confirming that our solution is correct.

Key Concepts

Exponent Rules

Exponent rules are crucial tools that help us manipulate expressions involving powers, ultimately simplifying many types of mathematical problems. When dealing with exponents, several important laws are typically used:

• Product of Powers Rule: If you multiply like bases, add the exponents: \( a^m \times a^n = a^{m+n} \).
• Quotient of Powers Rule: If you divide like bases, subtract the exponents: \( a^m \div a^n = a^{m-n} \).
• Power of a Power Rule: Raise a power to another power by multiplying the exponents: \( (a^m)^n = a^{m \times n} \).
• Negative Exponent Rule: A negative exponent denotes the reciprocal: \( a^{-n} = \frac{1}{a^n} \).
• Zero Exponent Rule: Any base with an exponent of zero is equal to one: \( a^0 = 1 \).

For our example, understanding the negative exponent rule was key because it allowed us to rewrite \( 0.1 \) as \( 10^{-1} \). This simplification directly led to solving the equation by equating the exponents.

Logarithms

Logarithms are the inverse operations of exponentiation, which are essential for solving equations where the unknown variable appears as an exponent. If you have an expression of the form \( b^x = y \), you can find \( x \) by converting this into a logarithmic form: \( \log_b(y) = x \).
The logarithm functions have several properties that make them powerful tools for solving exponential equations:

• Product Rule: \( \log_b(MN) = \log_b(M) + \log_b(N) \)
• Quotient Rule: \( \log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N) \)
• Power Rule: \( \log_b(M^n) = n \cdot \log_b(M) \)
• Change of Base Formula: \( \log_b(M) = \frac{\log_k(M)}{\log_k(b)} \)

By converting an equation to its logarithmic form, you can more easily isolate the variable and find solutions.

Equation Solving

Equation solving involves a variety of techniques, depending on the type of equation and its complexity. For exponential equations like \( 10^x = 0.1 \), recognizing equivalent base forms significantly simplifies the problem.
Here's a general approach to solving exponential equations:

• Identify common bases: Rewrite terms as powers of the same base, if possible.
• Use Logarithms: Apply logarithms to both sides when bases cannot be easily reconciled. This is particularly useful when the exponent is the unknown variable.
• Solve for the variable: Once you've reconciled or simplified the equation, solve for the variable using basic algebraic principles.

In our example, rewriting \( 0.1 \) as \( 10^{-1} \) allowed us to equate the exponents because both sides of the equation had the same base.

Base 10 Logarithms

Base 10 logarithms, often referred to as common logarithms, use the base of 10. They are simply expressed as \( \log(x) \) rather than \( \log_{10}(x) \). Common logarithms are particularly useful in scientific and engineering contexts for simplifying multiplicative relationships.
Why are base 10 logarithms so handy?

• Simplification: When solving equations where the base is 10, common logarithms directly simplify calculations involving exponents.
• Accessibility: Most scientific calculators easily handle base 10 logarithmic calculations.
• Relevance: They relate to the orders of magnitude, as in the Richter scale or pH levels.

In situations like our original equation, recognizing the value of using base 10 logarithms can make solving for unknowns straightforward, especially when we deal with decimal numbers equivalent to powers of 10.