Problem 80 - Algebra 2

Question

Solve each equation. If necessary, round to the nearest ten-thousandth. $$ 14^{9 x}=146 $$

Step-by-Step Solution

Answer

After solving, x should be approximately \(x ~= \frac{log(146)}{9*log(14)}\), depending on how much it needs to be rounded.

Step 1: Convert exponential form to logarithmic form

An exponential equation in the form x^a = b can be written in logarithmic form as log_x(b) = a. So, our equation \(14^{9 x}=146\) can be written as \(log_{14}(146) = 9x\)

Step 2: Isolate the variable x

To isolate x, divide both sides of the equation by 9, yielding: \(x = \frac{log_{14}(146)}{9}\)

Step 3: Calculate the value

Now we can compute the value of x. Using the logarithm base change rule, log_b(a) = log(a)/log(b), we see that \(x ~= \frac{log(146)}{9*log(14)}\). We can round our answer to the nearest ten-thousandth, if necessary.

Key Concepts

Logarithmic Functions

Logarithmic functions are the inverse operations of exponential functions, much like subtraction is the inverse of addition. In exponential equations, such as \(14^{9x} = 146\), logarithms help us "undo" the exponent to solve for the unknown variable. A logarithm answers the question: To what power must we raise a given base to get a specific number? In this context, if we have an equation in the form \(a^x = b\), it can be transformed into a logarithmic form as \(\log_a(b) = x\). This transformation is particularly useful when dealing with exponential equations because it allows us to work with simpler, linear forms by eliminating the exponent.
When solving our example, converting the exponential form \(14^{9x} = 146\) into its logarithmic counterpart gives us \(\log_{14}(146) = 9x\). This step brings the equation into a more manageable form, making it easier to isolate and solve for the variable \(x\). With the logarithmic expression, we can now use other algebraic techniques to continue solving the equation.

• Logarithmic functions are handy for simplifying exponential equations.
• They answer the question of which power a base must be raised to produce a given number.
• Logarithmic transformation is a crucial step for solving equations with exponents.

Equation Solving

Solving equations involves finding the value of unknown variables that make the equation valid. In our case, we want to solve for \(x\) in the equation \(14^{9x} = 146\).
We initially rewrite this exponential equation in logarithmic form, as seen in the previous section, to simplify the task. The critical step in solving the equation is to isolate the variable \(x\). After converting to logarithmic form \(\log_{14}(146) = 9x\), we can perform algebraic manipulations to separate \(x\) on one side. By dividing both sides of the equation by 9, we isolate \(x\), obtaining the expression \(x = \frac{\log_{14}(146)}{9}\). This step-by-step approach ensures clarity and eliminates potential sources of error.

• Rewrite equations in a more manageable form if necessary.
• Always isolate the variable using algebraic operations.
• Verify your results to ensure they satisfy the original equation.

Base Change Rule

The base change rule is an essential tool when working with logarithms, especially when you don’t have a calculator that supports direct computation of logarithms with arbitrary bases. The rule states that for any logarithm \(\log_b(a)\), you can change it to another base using the formula \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\).
This is particularly handy in our solution since most calculators and software naturally support logarithms of base 10 or \(e\) (natural logarithms). By applying the base change rule to the expression \(\log_{14}(146)\), we get \(\log_{14}(146) = \frac{\log(146)}{\log(14)}\). Utilizing this rule helps simplify the calculations and complete the solution with precision.
In our problem, this conversion allowed us to calculate \(x\) accurately as \(x = \frac{\log(146)}{9 \cdot \log(14)}\). This approach is fundamental when you need to switch between different log bases for practical computation.

• The base change rule simplifies calculations by converting logs to standard bases.
• Common bases include 10 (decimal) and \(e\) (natural logarithms).
• This rule is especially helpful when direct calculation is restricted.