Problem 5 - Algebra 2

Question

Solve each equation. Round to four decimal places. $$ 9^{x}=45 $$

Step-by-Step Solution

Answer

The solution is \(x \approx 1.7321\).

Step 1: Take Logarithms on Both Sides

Start by taking the logarithm of both sides of the equation to help isolate the variable, using the property that allows us to bring down the exponent: \[\log(9^x) = \log(45).\]

Step 2: Apply Logarithmic Identities

Use the logarithmic identity \( \log(a^b) = b\cdot\log(a) \) to bring down the exponent:\[x \cdot \log(9) = \log(45).\]

Step 3: Isolate the Variable \'x\'

To solve for \(x\), divide both sides of the equation by \(\log(9)\):\[x = \frac{\log(45)}{\log(9)}.\]

Step 4: Calculate the Logarithms and the Quotient

Use a calculator to find the values of the logarithms and then compute the solution:\[\log(45) \approx 1.6532,\]\[\log(9) \approx 0.9542,\]\[x \approx \frac{1.6532}{0.9542} \approx 1.7321.\]

Step 5: Round the Result

Round the result of the calculation to four decimal places if necessary. The result is already in four decimal places:\[x \approx 1.7321.\]

Key Concepts

Logarithms

Logarithms are a powerful mathematical tool that help us solve equations involving exponents. At their core, logarithms answer the question, "To what exponent must we raise a base number to get a given value?" For example, in the problem \(9^x = 45\), we want to find \(x\) for which raising 9 results in 45. By using logarithms, we transform multiplication and division problems with exponents into addition and subtraction problems, which are often easier to solve.

The notation \(\log_b(a)\) means "the logarithm of \(a\) with base \(b\)." In our problem, because we use common logarithms (base 10), we write \(\log(9^x)\) and \(\log(45)\), removing the need to specify the base. Using logarithms to solve equations like these is particularly useful when dealing with non-integer solutions.

Exponent Properties

Understanding exponent properties is key to solving logarithmic equations. One fundamental property is the power rule for logarithms: \(\log(a^b) = b \cdot \log(a)\). This allows us to "bring down" exponents for easier manipulation. In our exercise, we applied this rule to \(\log(9^x)\), converting it into \(x \cdot \log(9)\). This step effectively transforms the exponential equation into a linear form which is much simpler to solve.

Other important exponent properties include:

• Multiplication: \(a^m \cdot a^n = a^{m+n}\)
• Division: \(\frac{a^m}{a^n} = a^{m-n}\)
• Power to a power: \((a^m)^n = a^{m\cdot n}\)

Understanding these properties enables us to manipulate expressions and solve complex equations efficiently.

Solving Equations

Solving equations like \(9^x = 45\) involves a systematic approach. First, we transform the equation by taking the logarithm of both sides. This makes it easier to handle problems with exponents. Using the property \(\log(a^b) = b \cdot \log(a)\), we can bring down the exponent \(x\).

The next step is isolating the variable, which means getting \(x\) by itself. This is achieved by dividing both sides by \(\log(9)\), giving us \(x = \frac{\log(45)}{\log(9)}\). Solving equations often involves isolating the variable through addition, subtraction, multiplication, or division, depending on the operation present.

Calculating the solution requires evaluating the logarithms, which can be done using a calculator. Finally, complete the quotient to solve for \(x\). This systematic method is crucial for solving equations effectively.

Rounding Numbers

Rounding numbers is an essential skill, especially in mathematics and science where precision is crucial. When we solved the equation \(9^x = 45\), the result of \(x \approx 1.7321\) was rounded to four decimal places as part of providing an accurate solution.

Rounding helps simplify complex numbers, making them easier to work with while maintaining reasonable accuracy. Here’s a quick guide to rounding:

• Identify the place you need to round to.
• Look at the digit immediately to the right of this place.
• If it's 5 or more, round up; if it's less than 5, round down.

Rounding ensures that solutions are presented in a standard format, especially when exact decimals are not necessary or when tools only provide limited precision.