Please help, I need help on these two problems.

Solve the equation.
log3(x − 16) = log3(9x)

e^1 + ln(7)

It appears on the homework that the ln(7) is written like this ^ln(7) but I am still stuck on it.

Thanks!

if logs are the same, the arguments are the same, so

x-16 = 9x
x = -2
But, log(x) is not defined for x<0, regardless of the base

if you mean the entire exponents is 1+ln(7), then

e^(1+ln(7)) = e^1 * e^ln(7) = 7e

by definition, e^ln(n) = n

Log3 15

Sure, I can help you with these two problems.

1. Solve the equation log3(x - 16) = log3(9x):

To solve this equation, you can use the fact that if two logarithms have the same base, then their arguments must be equal. In this case, both logarithms have base 3, so the arguments must be equal.

So, we can set the arguments equal to each other:
x - 16 = 9x

Now, solve for x:
x - 9x = 16
-8x = 16
x = -2

So the solution to the equation is x = -2.

2. Evaluate e^1 + ln(7):

To further understand this, let's break it down step by step.

First, let's clarify what each term means:
- e represents the mathematical constant approximately equal to 2.71828.
- ln(7) represents the natural logarithm of 7.

Now, let's evaluate each term separately:

- e^1: This means raising the mathematical constant e to the power of 1. Raising any number to the power of 1 simply gives you the number itself. Therefore, e^1 is equal to e.

- ln(7): The natural logarithm of a number gives you the exponent that e needs to be raised to in order to get that number. In this case, ln(7) represents the exponent you need to raise e to in order to get 7. However, it seems there is a typo in your question with the formatting "^ln(7)". Disregard that formatting, and simply consider ln(7) as the natural logarithm of 7.

So, to evaluate e^1 + ln(7), substitute the above values:
e^1 + ln(7) = e + ln(7)

And since we don't know the exact value of ln(7), we can't simplify it further unless we use a calculator or approximation techniques.

Therefore, the final expression is e + ln(7), which is the sum of the mathematical constant e and the natural logarithm of 7.