Can you please help me with the following log. problems? thanxs!

Solve the exponential equation. Use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution.

7^x = 6^(x + 7)
this is how far I got: x log 7/x log 7= x+7 log 6/log 7, I am not too sure what to do next.

Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer.

log9 11 + log9 x = 1 ( I don't understand what to do)

log3(x + 5) = 2 + log3(x - 1) (For this problem I know I have to combine 2(x-1), but I don't know what to do next)

log10(x - 30) = 3 - log10 x (For this problem, I thought about solving it like x-30=3x, but I tried that I didn't get the answer I was suppose to)

Please limit yourself to one question per post. Here is how to do two of them

2. log9 11 + log9 x = 1

(log9 means log to base 9)

log9 (11x) = 1 = log9 9
(The log to any base of the base itself is 1)

11x = 9

x = 9/11

3. log3(x + 5) = 2 + log3(x - 1)
log3 [(x+5)/(x-1)] = 2 = log3 (9)
(since 3^2 = 9, log3 9 = 2)
(x+5)/(x-1) = 9
x+5 = 9x - 9
Now solve for 9

Sure! I can help you with those problems. Let's solve them step-by-step:

1. Solve the exponential equation: 7^x = 6^(x + 7)

To solve this equation, you can take the logarithm of both sides using any base you prefer. In this case, let's use the natural logarithm (log base e) for simplicity.

ln(7^x) = ln(6^(x + 7))

Using the logarithm property (ln(a^b) = b * ln(a)):

x * ln(7) = (x + 7) * ln(6)

Distribute ln(6) on the right side:

x * ln(7) = x * ln(6) + 7 * ln(6)

Move all terms involving x to one side:

x * ln(7) - x * ln(6) = 7 * ln(6)

Factor out x:

x * (ln(7) - ln(6)) = 7 * ln(6)

Divide both sides by (ln(7) - ln(6)):

x = (7 * ln(6)) / (ln(7) - ln(6))

Now you can use a calculator to evaluate the right side of the equation to obtain a decimal approximation. Remember to round your answer to two decimal places.

2. Solve the logarithmic equation: log9(11) + log9(x) = 1

To solve this equation, you can combine the logarithms using the logarithm property (log(a) + log(b) = log(ab)):

log9(11x) = 1

Since the logarithm is base 9, we can rewrite 1 as log9(9):

log9(11x) = log9(9)

Now we can equate the arguments:

11x = 9

Solve for x:

x = 9/11 (exact answer)

3. Solve the logarithmic equation: log3(x + 5) = 2 + log3(x - 1)

To solve this equation, let's first simplify by combining the logarithms on the right side:

log3(x + 5) - log3(x - 1) = 2

Using the logarithm property (log(a) - log(b) = log(a/b)), we can rewrite the equation as:

log3((x + 5)/(x - 1)) = 2

Now we can raise both sides to the power of 3:

3^log3((x + 5)/(x - 1)) = 3^2

Simplifying the left side:

(x+5)/(x-1) = 9

Cross-multiply:

9(x - 1) = (x + 5)

Solve for x:

9x - 9 = x + 5
8x = 14
x = 7/4 = 1.75 (decimal approximation)

4. Solve the logarithmic equation: log10(x - 30) = 3 - log10(x)

To solve this equation, let's first simplify by moving the terms involving logarithms to one side:

log10(x - 30) + log10(x) = 3

Using the logarithm property (log(a) + log(b) = log(ab)):

log10[(x - 30) * x] = 3

Combine the terms inside the logarithm:

log10(x^2 - 30x) = 3

Now, rewrite the equation in exponential form:

10^3 = x^2 - 30x

Simplify:

1000 = x^2 - 30x

Rearrange the equation as a quadratic equation:

x^2 - 30x - 1000 = 0

Now you can solve this quadratic equation using factoring, completing the square, or the quadratic formula. The solutions are x = -10 and x = 40.

However, we need to consider the domain of the original logarithmic expressions. For log10(x - 30) and log10(x), the argument (x - 30) and x must be greater than 0. Therefore, we reject x = -10 as it does not satisfy this condition.

Therefore, the only valid solution is x = 40.

Sure, I can help you with these logarithmic and exponential equations. Let's solve each one step by step.

1. Exponential Equation: 7^x = 6^(x + 7)

To solve this equation, we need to use logarithms. Let's take the logarithm of both sides of the equation. Since the bases are different (7 and 6), we can choose any base for the logarithm. Let's use the natural logarithm (ln) for this example:

ln(7^x) = ln(6^(x + 7))

Now, we can apply the power rule of logarithms:

x * ln(7) = (x + 7) * ln(6)

Next, distribute the ln(6) on the right side:

x * ln(7) = x * ln(6) + 7 * ln(6)

To isolate the variable x, we can subtract x * ln(6) from both sides:

x * ln(7) - x * ln(6) = 7 * ln(6)

Now, factor out x on the left side:

x * (ln(7) - ln(6)) = 7 * ln(6)

Finally, divide both sides by (ln(7) - ln(6)) to solve for x:

x = (7 * ln(6)) / (ln(7) - ln(6))

Using a calculator, you can approximate the value of x to two decimal places.

2. Logarithmic Equation: log9 11 + log9 x = 1

To solve this logarithmic equation, we can apply the logarithmic properties.

Since we have the sum of logarithms on the left side, we can use the product rule of logarithms:

log9(11x) = 1

Now, we can rewrite the equation in exponential form:

9^1 = 11x

Simplifying, we find x = 1/11.

Note: We need to reject any value of x that is not in the domain of the original logarithmic expression. In this case, x = 1/11 is the only valid solution.

3. Logarithmic Equation: log3(x + 5) = 2 + log3(x - 1)

To solve this logarithmic equation, we can use the properties of logarithms.

First, let's simplify the equation by combining the logarithms on the right side:

log3(x + 5) - log3(x - 1) = 2

Using the quotient rule of logarithms, we can rewrite the left side:

log3((x + 5)/(x - 1)) = 2

Now, we can rewrite the equation in exponential form:

3^2 = (x + 5)/(x - 1)

Solving for x, we find:

9(x - 1) = x + 5

Simplifying further:

9x - 9 = x + 5

8x = 14

x = 7/4

So, the exact solution is x = 7/4.

4. Logarithmic Equation: log10(x - 30) = 3 - log10(x)

To solve this logarithmic equation, we can rewrite the equation using the properties of logarithms.

First, let's combine the logarithms on the right side using the quotient rule:

log10(x - 30) = 3 + log10(1/x)

Now, we can take the antilog of both sides to eliminate the logarithms:

x - 30 = 10^3 * (1/x)

Simplifying, we have:

x - 30 = 1000/x

Multiplying both sides by x to eliminate the fraction:

x(x - 30) = 1000

x^2 - 30x - 1000 = 0

Solving this quadratic equation, we find two solutions: x = -25 and x = 40.

However, we need to check the solutions and reject any values that are not in the domain of the original logarithmic expressions. In this case, x = -25 is not valid because we cannot take the logarithm of a negative number. Therefore, the only valid solution is x = 40.

I hope this helps! Let me know if you have any further questions.