Solving Exponential Equations: Find X In 69 * 2^x = 621

Let's dive into solving an exponential equation! Specifically, we're going to tackle the equation $69 umber_multiplication 2^x = 621$ and find the value of $x$, rounding our answer to the nearest hundredth. This is a classic problem in algebra that combines the concepts of exponential functions and logarithms. Buckle up, guys, because we're about to embark on a mathematical journey!

Step-by-Step Solution

To solve for $x$ in the equation $69 umber_multiplication 2^x = 621$, we'll follow these steps:

Step 1: Isolate the Exponential Term

First, we need to isolate the exponential term, which in this case is $2^x$. To do this, we'll divide both sides of the equation by 69:

$2^x = \frac{621}{69}$

Simplifying the fraction, we get:

$2^x = 9$

Step 2: Apply Logarithms

Now that we have isolated the exponential term, we can apply logarithms to both sides of the equation. The logarithm allows us to bring the exponent $x$ down as a coefficient. We can use any base for the logarithm, but the common logarithm (base 10) or the natural logarithm (base $e$) are often the most convenient. Let's use the natural logarithm (ln) for this example:

$\ln(2^x) = \ln(9)$

Using the power rule of logarithms, which states that $\ln(a^b) = b \number_multiplication \ln(a)$, we can rewrite the left side of the equation:

$x \number_multiplication \ln(2) = \ln(9)$

Step 3: Solve for $x$

To solve for $x$, we'll divide both sides of the equation by $\ln(2)$:

$x = \frac{\ln(9)}{\ln(2)}$

Now, we can use a calculator to find the values of $\ln(9)$ and $\ln(2)$:

$\ln(9) \approx 2.1972$

$\ln(2) \approx 0.6931$

So,

$x \approx \frac{2.1972}{0.6931} \approx 3.1699$

Step 4: Round to the Nearest Hundredth

Finally, we round our answer to the nearest hundredth:

$x \approx 3.17$

So, the solution to the equation $69 \number_multiplication 2^x = 621$, rounded to the nearest hundredth, is approximately $x = 3.17$.

Alternative Method: Using Base 2 Logarithm

Another approach to solving this equation involves using the base 2 logarithm, denoted as $\log_2$. This can simplify the process slightly.

Step 1: Isolate the Exponential Term (Same as Before)

$2^x = \frac{621}{69} = 9$

Step 2: Apply Base 2 Logarithm

Apply the base 2 logarithm to both sides of the equation:

$\log_2(2^x) = \log_2(9)$

Using the property that $\log_b(b^x) = x$, we get:

$x = \log_2(9)$

Step 3: Use Calculator or Change of Base Formula

Most calculators do not have a direct way to calculate $\log_2(9)$. However, we can use the change of base formula to convert it to a logarithm with a more common base (like base 10 or base $e$):

$\log_2(9) = \frac{\log_{10}(9)}{\log_{10}(2)} = \frac{\ln(9)}{\ln(2)}$

This is the same expression we obtained earlier when using the natural logarithm. Therefore, the rest of the steps are the same.

Step 4: Calculate and Round

Using a calculator:

$x = \frac{\ln(9)}{\ln(2)} \approx \frac{2.1972}{0.6931} \approx 3.1699$

Rounding to the nearest hundredth:

$x \approx 3.17$

Key Concepts Used

Exponential Equations

Exponential equations are equations in which the variable appears in the exponent. These types of equations are common in various fields, including finance, physics, and engineering. Solving them often involves isolating the exponential term and using logarithms.

Logarithms

Logarithms are the inverse operation to exponentiation. The logarithm of a number $x$ with respect to a base $b$ is the exponent to which $b$ must be raised to produce $x$. In other words, if $b^y = x$, then $\log_b(x) = y$. Logarithms are essential for solving exponential equations because they allow us to "undo" the exponentiation.

Properties of Logarithms

Several properties of logarithms are useful when solving equations:

1. Power Rule: $\ln(a^b) = b \number_multiplication \ln(a)$ (or $\log_b(a^c) = c \number_multiplication \log_b(a)$)
2. Change of Base Formula: $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$

Natural Logarithm

The natural logarithm, denoted as $\ln(x)$, is the logarithm to the base $e$, where $e$ is an irrational number approximately equal to 2.71828. The natural logarithm is widely used in calculus and other areas of mathematics.

Common Mistakes to Avoid

1. Incorrectly Applying Logarithms: Make sure to apply logarithms to both sides of the equation to maintain equality. Also, be careful when using logarithm properties.
2. Rounding Errors: Avoid rounding intermediate values, as this can lead to inaccuracies in the final answer. Keep as many decimal places as possible until the final step.
3. Forgetting to Isolate the Exponential Term: Always isolate the exponential term before applying logarithms. This simplifies the equation and makes it easier to solve.
4. Misunderstanding Logarithm Properties: Ensure you understand and correctly apply the properties of logarithms, such as the power rule and the change of base formula.

Real-World Applications

Compound Interest

Exponential equations are frequently used in finance to calculate compound interest. For example, the formula for compound interest is:

$A = P(1 + \frac{r}{n})^{nt}$

Where:

• $A$ is the amount of money accumulated after n years, including interest.
• $P$ is the principal amount (the initial amount of money).
• $r$ is the annual interest rate (as a decimal).
• $n$ is the number of times that interest is compounded per year.
• $t$ is the number of years the money is invested or borrowed for.

Solving for $t$ in this equation often involves using logarithms.

Population Growth

Population growth can often be modeled using exponential equations. The formula for exponential growth is:

$N(t) = N_0 e^{kt}$

Where:

• $N(t)$ is the population at time $t$.
• $N_0$ is the initial population.
• $k$ is the growth rate.
• $t$ is the time.

Radioactive Decay

In physics, radioactive decay is modeled using exponential equations. The amount of a radioactive substance remaining after time $t$ is given by:

$N(t) = N_0 e^{-\lambda t}$

Where:

• $N(t)$ is the amount of substance remaining at time $t$.
• $N_0$ is the initial amount of the substance.
• $\lambda$ is the decay constant.
• $t$ is the time.

Practice Problems

1. Solve for $x$: $15 \number_multiplication 3^x = 405$
2. Solve for $x$: $7 \number_multiplication 5^x = 875$
3. Solve for $x$: $2 \number_multiplication e^x = 10$
4. Solve for $x$: $10 \number_multiplication 2^x = 160$

Conclusion

Solving exponential equations like $69 \number_multiplication 2^x = 621$ involves isolating the exponential term, applying logarithms, and using logarithm properties to solve for the variable. Remember to round your answer to the specified degree of accuracy. By understanding these steps and practicing regularly, you'll become more confident in solving exponential equations. Keep up the great work, and happy solving!