I am having a very hard time with these...I am very confused and unsure of where to start with these...if someone could please guide me through one or two of them to help me understand, i would greatly appreciate it....
hello,
to #9:
use the base 8 at both sides of the equation:
$\displaystyle 8^{\log_8{(4x+14)}}=8^2~\implies~4x+14=64$ . Please continue.
to #10:
$\displaystyle 6 \cdot e^{0.05 \cdot x} + 29 = 83~\iff~e^{0.05 \cdot x}=9~\implies~0.05 \cdot x = \ln(9)$ . Please continue.
to #11:
$\displaystyle \frac{400}{1+e^{-x}}=375~\iff~400=375+375 \cdot e^{-x}~\iff~\frac1{15}=\frac1{e^x}~\implies~e^x=15$ . Please continue.
to #12:
$\displaystyle \log_7\left(\frac{x^4}{\sqrt[6]{y}} \right)=\log_7\left(x^4 \cdot y^{-\frac16} \right)=4\cdot \log_7(x)-\frac16 \cdot \log_7(y)$
to #13:
use the property: $\displaystyle \log(a) + \log(b) = \log(a \cdot b)$
to #14:
use the property: $\displaystyle \log(a) - \log(b) = \log\left(\frac ab\right)$
to #15:
you are supposed to know: $\displaystyle \log_b(a)=\frac{\ln(a)}{\ln(b)}=\frac{\log_{10}(a) }{\log_{10}(b)}$
It doesn't matter which logarithmetic function you use. You should come out with
$\displaystyle \log_8(155) \approx 2.425374802...$
Hi,
logarithm is only another word for exponent. The base to which this exponent belongs is added as an index. Thus $\displaystyle \log_8()$ belongs to the base 8.
If you use this base and the logarithm then the result of this power is the value or term in the bracket following the word "log":
$\displaystyle 8^{\log_8 (4x+14)} = 4x+14$
In problem 11 you have
$\displaystyle 400 = 375(1 + e^{0.5x})$
Continuing:
$\displaystyle \frac{400}{375} = 1 + 3^{0.5x}$
(You left the 1 out on the RHS.)
Problem 12: What's this under the answer with the $\displaystyle 3x^0$ etc. It is unrelated to your problem. (Which you got right, by the way. )
Problem 15: The change of base formula is
$\displaystyle log_a(b) = \frac{log_c(b)}{log_c(a)}$
So
$\displaystyle log_8(155) = \frac{log_{10}(155)}{log_{10}(8)}$
This is not the same as
$\displaystyle log_{10} \left ( \frac{155}{8} \right )$
-Dan