plzzz help me with these math problems!!!!?

Question

Use the change of base rule to find the logarithm rounded to four decimal places.

log9 61.31


Solve the equation and express the solution in exact form.

log4 (x - 4) + log4 (x - 4) = 1

Answer 1

For the first question, a reminder that the change of base formula goes as follows:

log[base c](a) = {log[base b](a)} / {log[base b](c)}

Translation: You can turn a single log into a quotient of logs by choosing whatever base you wish.

Since we want the base of our log to be calculator friendly, you can choose b = 10. Then

Log [base 9] (61.31) = [log (61.31)] / [log (9)]

Another calculator friendly base would be e, resulting in taking the natural log, ln.

Log [base 9] (61.31) = ln (61.31) / ln (9)

To solve the equation

log[base 4](x-4) + log[base 4](x-4) = 1

First note that those two terms are exactly the same. So we can combine them since they are like terms.

2 log [base 4](x - 4) = 1

A reminder that log[base b](a^c) = c * log[base b](a). Therefore,

log[base4](x-4)^2 = 1

Now, we convert this to exponential form.

4^1 = (x-4)^2
4 = x^2 -8x + 16
x^2 - 8x + 12 = 0

Therefore, (x - 6)(x - 2) = 0
So x = 6 or x = 2
BUT WAIT! You can't assume those values will work; you have to test the values in the original function, and if you end up taking the log of a negative number, you discard it.

Test x = 6: log[base 4] (6 - 4) + log[base4](6-4) checks out.
Test x = 2: log[base 4] (2 - 4) ..... this one already fails.

Therefore, x = 6 is the only answer.

Answer 2

most calculators or available logarithm tables only provide base e or base 10 logarithms. so use the change of base rule

to calculate the base b log of any number x using base 10 logs:

logb x = (log10 x) / (log10 b)

for the given problem, this is:
log9 61.31 = log10 61.31 / log10 9 =
1.7875 / 0.9542 = 1.8732

to verify, raise 9 to the power of 1.8732

-------------
1 apple + 1 apple = 2 apples, so

log4(x-4) + log4(x-4) = 2log4(x-4) = 1
divide both sides by 2, then take the base 4 antilog (raise 4 to the power of each side)

x-4 = 4^(1/2) = +/- 2
add 4 to both sides
x = 4+/-2
x = 6, x = 2

however x=2 is an invalid answer to the original equation if
the log function is restricted to the reals

Answer 3

The change of bases rule is:
log[b] (n) = log(n) / log(b)

So figure:
log(61.31) / log(9)
≈ 1.78753132 / 0.954242509
≈ 1.87324637
which gets rounded to:
1.8732

PART 2:

Original equation:
log[4] (x - 4) + log[4] ( x - 4) = 1

Add the logs which are exactly the same:
2 log[4] (x - 4) = 1

Divide both sides by 2:
log[4] (x - 4) = 1/2

Now raise both sides upon the base 4:
x - 4 = 4^(1/2)

4^(1/2) is the same as sqrt(4) so the answers are ±2
x - 4 = ±2

Add 4 to both sides:
x = 4 ± 2

Simplify:
x = 2 or x = 6

However, you never take the log of a negative number, so 2 isn't a valid answer. Therefore x = 6.

Answer 4

Change of base can work for the first one, but here's another way that I like to do it:

Set log(base 9)[61.31] = x

Change from log form to exponential form: 9 ^ x = 61.31

Take the log of both sides: log[9 ^ x] = log[61.31]

Bring down the x via a log property: x * log[9] = log[61.31]

And divide both sides by log[9]: x = log[61.31] / log[9]

And so x is approximately 1.8733

Answer 5

First, we'll let e = the distance traveled by elephant and c = the distance traveled by chariot. We will also let x = the time spent on the elephant and y = the time spent on the chariot. Since neither the camp nor the outskirts of Rome are moving, we know the distance is constant. Therefore, e=c . Also, since Distance = Rate x Time (the "dirt" rule), we know that: e=2x and c=10y We also know that: x+y=18 If we set e and c equal, we get: 2x=10y, or y=x/5 (by dividing both sides by 10) If we set the second equation in standard form, we get y=18-x (by subtracting x from both sides) Now that we have y set to two expressions of x, we can set those expressions equal (this is known as substitution): x/5=18-x Multiply both sides by 5, and we get x=90-5x . Add 5x to both sides, and we get 6x=90 and x=15 (by dividing both sides by x). Take x all the way back and plug it into e=2x, and we end up with e=30. The distance from camp to the outskirts of Rome is 30 km. You can check your work by solving for y using x, which gives y=3. Plug 3 into the equation for c, and you get c=30=e.

Answer 6

log_9 61.31=x
61.31=9^x
ln 61.31=x ln 9
x=log_9 61.31=ln 61.31 / ln 9=1.8732

log_4 (x-4)+log_4(x-4)=1
2 log_4(x-4)-1
log_4(x-4)=.5
x-4=4^.5=2
x=6

Answer 7

not to positive about the first one but i think it's 58.5046
The second one is -3.6124