Solve Logarithmic Equations: Find The Missing Values

Hey there, math enthusiasts! Let's dive into some fun with logarithms. We've got three equations where we need to find the missing values to make them true. Don't worry; it's like a puzzle, and we'll solve it together! So, grab your thinking caps, and let's get started!

Equation 1: $\log _2 7+\log _2 11=\log _2 \square$

Logarithm Rules: When you're adding logs with the same base, you can combine them into a single log by multiplying the numbers inside.

Alright, guys, let's tackle our first equation: $\log _2 7+\log _2 11=\log _2 \square$. What we need to remember here is a fundamental property of logarithms: when you add two logarithms with the same base, you can combine them into a single logarithm by multiplying the arguments (the numbers inside the logarithm). In mathematical terms, this rule looks like this:

$\log_b{m} + \log_b{n} = \log_b{(m ") n}$

In our case, the base is 2, m is 7, and n is 11. So, we can rewrite the left side of the equation as:

$\log _2 7+\log _2 11 = \log _2 (7 \times 11)$

Now, we just need to multiply 7 and 11. Basic multiplication tells us that 7 times 11 equals 77. Therefore, our equation becomes:

$\log _2 7+\log _2 11 = \log _2 77$

So, the missing value that makes the equation true is 77. This is because $\log _2 77$ is the same as $\log _2 7+\log _2 11$. Remember, the key to solving these types of problems is recognizing and applying the logarithm rules correctly. In this case, we used the addition rule to combine two logarithms into one. And that's it! We've successfully found the missing value for the first equation. Now, let's move on to the next one and see what challenges await us there!

Equation 2: $\log _8 3-\log _8 \square=\log _8 \frac{3}{5}$

Logarithm Rules: When subtracting logs with the same base, you divide the numbers inside.

Okay, let's jump into the second equation: $\log _8 3-\log _8 \square=\log _8 \frac{3}{5}$. This time, we're dealing with subtraction of logarithms, which also has a handy rule. When you subtract two logarithms with the same base, you can combine them into a single logarithm by dividing the arguments. The rule looks like this:

$\log_b{m} - \log_b{n} = \log_b{\frac{m}{n}}$

In our equation, the base is 8, m is 3, and we're trying to find n. We want to find the value that, when we divide 3 by it, we get $\frac{3}{5}$. So, we can set up the equation like this:

$\log _8 3-\log _8 x = \log _8 \frac{3}{5}$

Using the subtraction rule, we can rewrite the left side as:

$\log _8 \frac{3}{x} = \log _8 \frac{3}{5}$

Now, for the two sides to be equal, the arguments inside the logarithms must be equal. That means:

$\frac{3}{x} = \frac{3}{5}$

To solve for x, we can cross-multiply:

$3 \times 5 = 3 \times x$

$15 = 3x$

Now, divide both sides by 3:

$x = \frac{15}{3}$

$x = 5$

So, the missing value is 5. This makes the equation true because $\log _8 3-\log _8 5$ is the same as $\log _8 \frac{3}{5}$. Remember, when you subtract logarithms, you divide the arguments, and when you add them, you multiply the arguments. Keeping these rules straight is super helpful for solving these kinds of problems. Great job, guys! We've conquered another equation. Only one more to go!

Equation 3: $\log _5 4=\square \log _5 2$

Logarithm Rules: A number multiplying a log can become the exponent of the number inside the log.

Last but not least, let's tackle the third equation: $\log _5 4=\square \log _5 2$. This one involves a different logarithm property, the power rule. The power rule states that if you have a logarithm multiplied by a number, you can move that number as an exponent of the argument inside the logarithm. In mathematical terms:

$a \log_b{m} = \log_b{m^a}$

In our equation, we have $\log _5 4$ on one side and $\square \log _5 2$ on the other. We want to find the number that, when placed in the square, makes both sides equal. Let's call that number a:

$\log _5 4 = a \log _5 2$

Using the power rule, we can rewrite the right side as:

$\log _5 4 = \log _5 2^a$

Now, for both sides to be equal, the arguments inside the logarithms must be equal. So, we have:

$4 = 2^a$

We need to find what power of 2 equals 4. We know that 2 squared (2 to the power of 2) is 4:

$4 = 2^2$

Therefore, a must be 2. So, our equation becomes:

$\log _5 4 = 2 \log _5 2$

This is true because $\log _5 4$ is the same as $\log _5 2^2$, which simplifies to $\log _5 4$. So, the missing value is 2. We've successfully used the power rule to find the missing value. Remember, the power rule is super useful when you need to change the coefficient of a logarithm into an exponent, or vice versa. Nice work, everyone! We've solved all three equations and found all the missing values. Keep practicing, and you'll become a logarithm master in no time!

Final Answers:

• $\log _2 7+\log _2 11=\log _2 **77**$
• $\log _8 3-\log _8 **5**=\log _8 \frac{3}{5}$
• $\log _5 4= **2** \log _5 2$

Conclusion: Great job, guys! You've successfully filled in the missing values to make all the equations true. Keep up the excellent work, and you'll be a math whiz in no time!