Logarithm Help Please?

Question

I am not looking for exact answers just guidance on what to do and would greatly appreciate it:)

I have 2 equations that say Solve For X, but I can't figure out what I need to do

10^x=384
e^x=116

I don't know if we exactly learned how to do those yet...

Another problem says Determine the Value of x & Y if 2^y=8^x and 3^y=3^x+4


Thanks for helping!

Answer 1

a^y = x (exponential form)
y = log base a (x) (logarithm form)

10^x = 384

Whenever you get a variable in the exponent, you need to use a log to solve. If the base is 10, use log. If the base is e, use ln.

x = log base 10 (384)

Log base 10 = log

x = log 384

Use a calculator to get x = 2.584
-----------------------------------------

e^x = 116

Use ln since the base is e. ln = log base e

x = ln 116

Use a calculator to get x = 4.754.
-------------------------------------------------

2^y = 8^x
3^y = 3^(x + 4) << I'm assuming x + 4 is the power?

To solve these, you need to make the bases the same.
8 = 2^3

2^y = (2^3)^x
2^y = 2^3x
y = 3x

3^y = 3^ (x + 4)
y = x + 4

Now you have your two equations.
y = 3x
y = x + 4

Use substitution to set both equal to each other.

3x = x + 4
2x = 4
x = 2

Substitute this in the other equation to find y.

y = 3x
y = 3(2)
y = 6

(2 , 6)

Hope that helps!

Answer 2

10^x = 384

take the logarithm of both sides:
log( 10^x ) = log (384)
x log (10) = log (384)

note: if you use base-10 logartims,

x = log (384)

/*****/
e^x = 116
take the logarithm of both sides:
log (e^x) = log (116)

note: if you use base-e logartihms, ("natural log")
x = log(116)
/*****/
2^y=8^x

uhhh, same story
"take the logarithm of both sides:"

y log(2) = x log(8) = x log (2^3) = x * 3 log (2)
factor out "log(2)"
y = 3x
note: this indicates there's an infinite number of solutions for x,y

3^y=3^x+4 now becomes
3^(3x) = 3^(x) + 4

hmmmm, here I would hope that you made a typo in the
problem statement:

ie it would be "nicer" if
3^y = 3^(x+4)
in which case since y = 3x
we would have
3x = x+4
2x = 4
x = 2

IF 3^y = (3^x) + 4

this problem is waaaay past the level of difficulty of the other
two p[roblems so I suspect a "typo" as indicated above

good luck


last question .... ugly

Answer 3

You can take the natural logarithm of both sides of e^x=116 to obtain ln(e^x) = ln(116). By definition, ln(e^x) = x, so this simplifies to x = ln(116).

You can do a similar thing with the first one except use logarithm base 10 in place of the natural logarithm.