Pre-test 6 help

**Brooke** · Jul 13th 2006, 03:15 PM

questions are on attachment!!

**galactus** · Jul 13th 2006, 03:51 PM

#6. Condense the expression to the logarithmic of a single quanity.
[2log 5 (x+4)+7log 5 (x+7)]-1/2log 5 x

Hello Brooke:

Use the properties of logs.

log(a)+log(b)=log(ab)...[1]

log(a)-log(b)=log(a/b)...[2]

Rewrite as:

$log_{5}(x+4)^{2}+log_{5}(x+7)^{7}-log_{5}x^{\frac{1}{2}}$

Using [1]

$log_{5}((x+4)^{2}(x+7)^{7})-log_{5}x^{\frac{1}{2}}$

Using [2]

$log_{5}(\frac{(x+4)^{2}(x+7)^{7}}{\sqrt{x}})$

**Soroban** · Jul 13th 2006, 04:25 PM

Hello, Brooke!

1. Sketch the graph of the function: $f(x) \:= \:3 + e^x$

You're expected to be familiar with the graph of $y\:=\:e^x$ .

This is the same graph translated up 3 units.

2. Evaluate the expression without using a calculator: $\log_{128}2$

Let: $\log_{128} \:=\;x$

Rewrite in exponential form: . $128^x\:=\:2$

Get the same base on both sides: . $(2^7)^x\:=\:2\quad\Rightarrow\quad 2^{7x}\:=\:2^1$

Since the bases are equal, the exponents are equal: . $7x\:=\:1\quad\Rightarrow\quad x\,=\,\frac{1}{7}$

Therefore: . $\log_{128}2\;=\;\frac{1}{7}$

3. Identify the logarithmic equation written in exponential form: $\log_{243}81\:=\:\frac{4}{5}$

Not sure what "identify" means here . . .

In exponential form, the expression is: . $243^{\frac{4}{5}}\:=\:81$

4. Evaluate using the change-of-base formula: $\log_9517$

We have: . $\log_9517\:=\:\frac{\ln517}{\ln9}\:=\:2.843606857. ..$

5. Which is the logarithm rewritten as a ratio of natural logarithms?
. . . $A\!:\;\ln\left(\frac{3}{x}\right)\qquad B\!:\;\frac{\ln 3}{\ln x}\qquad C\!:$ $\;\ln x - \ln 3\qquad D\!:\;\frac{\ln x}{\ln 3}$

Is there a typo? . . . Both $B$ and $D$ are ratios of natural logs.

6. Condense the expression to the logarithm of a single quantity:
. . . $\left[2\log_5(x+4) + 7\log_5(x + 7)\right] - \frac{1}{2}\log_5x$

$\left[2\cdot\log_5(x+4) + 7\cdot\log_5(x+7)\right] - \frac{1}{2}\cdot\log_5(x)$

. . $= \;\left[\log_5(x+4)^2 + \log_5(x+7)^7\right] - \log_5\!\left(x^{\frac{1}{2}}\right)$

. . $= \;\log_5\!\left[(x+4)^2\cdot(x+7)^7\right] - \log_5\!\left(x^{\frac{1}{2}}\right)$

. . $= \;\log_5\!\left[\frac{(x+4)^2(x+7)^7}{x^{\frac{1}{2}}}\right]$

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