4. here is a table of some logarithm values.
a. what is the approximate value of $\log_{10}(400)$?
b. what is the value of $\log_{10}(1000)$? is this value approximate or exact? explain how you know.
5. what is the value of $\log_{10}(1,000,000,000)$? explain how you know.
6. a bank account balance, in dollars, is modeled by the equation $f(t) = 1,000 \cdot (1.08)^t$ where $t$ is time measured in years.
about how many years will it take for the account balance to double? explain or show how you know.

Question

Accepted Answer

### 4a
# Explanation:
## Step1: Recall logarithm table
From the given table, when $ x = 400 $, the logarithmic value (presumably $ \log_{10}(x) $) is provided as $ 2.6021 $ (as seen in the handwritten note and the table structure).
# Answer:
$ 2.6021 $

### 4b
# Explanation:
## Step1: Recall logarithm definition
The logarithm base $ 10 $ of a number $ y $, $ \log_{10}(y) $, is the exponent $ n $ such that $ 10^n = y $. For $ y = 1000 $, we know that $ 10^3=1000 $.
## Step2: Determine exactness
Since $ 10^3 = 1000 $ holds exactly (by the definition of exponents and logarithms), $ \log_{10}(1000)=3 $ is an exact value, not an approximation.
# Answer:
The value of $ \log_{10}(1000) $ is $ 3 $, and it is exact because $ 10^3 = 1000 $, so by the definition of a base - 10 logarithm, $ \log_{10}(1000)=3 $ exactly.

### 5
# Explanation:
## Step1: Express the number in exponential form
We can write $ 1,000,000,000 $ as $ 10^9 $ (since $ 10\times10\times\cdots\times10 $ (9 times) $ = 10^9=1,000,000,000 $).
## Step2: Apply logarithm definition
Using the definition of a logarithm, $ \log_{10}(a^b)=b\log_{10}(a) $. For $ a = 10 $ and $ b = 9 $, $ \log_{10}(10^9)=9\log_{10}(10) $. And since $ \log_{10}(10) = 1 $ (by the definition of a logarithm, as $ 10^1 = 10 $), we have $ \log_{10}(10^9)=9\times1 = 9 $.
# Answer:
The value of $ \log_{10}(1,000,000,000) $ is $ 9 $. We know this because $ 1,000,000,000=10^9 $, and by the definition of a base - 10 logarithm $ \log_{10}(10^n)=n $, so $ \log_{10}(10^9) = 9 $.

### 6 (Assuming the function is $ f(t)=1000\times(1.08)^t $ as it seems to be a common compound - interest - like model with an interest rate of 8%)
# Explanation:
## Step1: Set up the equation for doubling
We want to find $ t $ when $ f(t)=2\times1000 = 2000 $. So we set up the equation $ 2000=1000\times(1.08)^t $.
## Step2: Simplify the equation
Divide both sides of the equation by $ 1000 $: $ \frac{2000}{1000}=(1.08)^t $, which simplifies to $ 2=(1.08)^t $.
## Step3: Take logarithms on both sides
Take the base - 10 logarithm (or natural logarithm) of both sides. Using base - 10 logarithms: $ \log_{10}(2)=\log_{10}(1.08^t) $.
## Step4: Use logarithm power rule
By the power rule of logarithms $ \log_{a}(b^c)=c\log_{a}(b) $, so $ \log_{10}(2)=t\log_{10}(1.08) $.
## Step5: Solve for $ t $
We know that $ \log_{10}(2)\approx0.3010 $ and $ \log_{10}(1.08)\approx0.0334 $. Then $ t=\frac{\log_{10}(2)}{\log_{10}(1.08)}\approx\frac{0.3010}{0.0334}\approx9 $ (or using natural logarithms: $ \ln(2)=t\ln(1.08) $, $ t = \frac{\ln(2)}{\ln(1.08)}\approx\frac{0.6931}{0.07696}\approx9 $).
# Answer:
It will take about 9 years for the account balance to double. We set up the equation $ 2000 = 1000\times(1.08)^t $, simplify to $ 2=(1.08)^t $, take logarithms, use the power rule, and solve for $ t $ to get $ t\approx9 $.

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QUESTION IMAGE

Question

Explanation:

4a

Step1: Recall logarithm table

Step1: Recall logarithm definition

Step2: Determine exactness

Step1: Express the number in exponential form

Step2: Apply logarithm definition

Answer:

4b