Question

These are some I chose to practice but I have a hard time knowing how to do it. The answers are online but that doesn't help. These deal with logarithmic functions etc. PLEASE HELP! THANKYOU

1.)solve the equation:
3^(x+1)=27^(x+3)

2.) e^x=5
3.) 2^(3x)+9=25
4.) 4^(x+1)=21
5.) log6 (5x+8)=log6 (13x)

write an exponential function y=ab^x whose graph passes through the given points.

1.) (1,6), (2,36)
2.) (2,16), (3,64)

write a power function y=ax^b whose graph passes through the given points.

1.) (2,2), (4,16)
2.) (3,3), (6,12)

1.) A store begins selling a new type of baseball shoe. The table shows the number (y) of pairs sold during week (x). Find a power model for the following data.

week x| 1 | 2 |
pair y| 10 | 80 |

Answer 1

I'll solve for the first five problems. You have to recall some laws of exponents for these.
(1)
3^(x+1) = 27^(x+3)
Note that we can rewrite 27 as power of 3: 27 = 3^3. Thus,
3^(x+1) = (3^3)^(x+3)
You can distribute the '3' in 3^3 to (x+3).
3^(x+1) = 3^(3(x+3))
3^(x+1) = 3^(3x+9)
Now that they have the same base, we can equate their exponents:
x + 1 = 3x + 9
x - 3x = 9 - 1
-2x = 8
x = -4

(2)
e^x = 5
Get the natural logarithm (ln) of both sides:
ln (e^x) = ln (5)
Note that we can rewrite ln (e^x) into x*ln(e). But ln(e) is equal to 1. So,
x * 1 = ln(5)
x = ln(5)

(3)
2^(3x) + 9 = 25
2^(3x) = 25 - 9
2^(3x) = 16
This is almost similar to #1. We rewrite 16 as power of 2: 16 = 2^4. So,
2^(3x) = 2^4
I believe you know what to do next. ;)

(4)
4^(x+1) = 21
Now, this doesn't have an exact answer. What we'll do is get the ln of both sides:
ln (4^(x+1)) = ln (21)
Again, for ln (4^(x+1)), we can put the exponent outside and leave the base inside, so ln (4^(x+1)) = (x+1)*ln(4).
(x+1) * ln(4) = ln(21)
x + 1 = ln(21) / ln(4)
x = [ ln(21) / ln(4) ] - 1

(5)
log6 (5x+8) = log6 (13x)
If that means "base 6", then simply equate the terms inside the log:
5x + 8 = 13x
5x - 13x = -8
-8x = -8
x = 1

hope this helps~ `u`

Answer 2

For the next questions, they are all solved with the same method. So I'll guide you to solving #1, then try solving the others by following these steps.

We need to find an equation in the form y = ab^x that passes through the points (1,6) and (2,36).
What we'll do is substitute each point to the equation, and solve for 'a' and 'b'
For (1,6):
y = ab^x
6 = ab^1
6 = ab

For (2,36):
y = ab^x
36 = ab^2

From the first equation, (6 = ab), we can have an expression for a, such that:
a = 6 / b
We can substitute this into the second equation:
36 = ab^2
36 = (6 / b) * b^2
36 = 6b^2 / b
36 = 6b
b = 36/6
b = 6

Now that we have a value for b, we can substitute this to first equation to get a:
6 = a(6)
a = 6/6
a = 1

Therefore, your equation is
y = 6^x

hope this helps~ `u`