9. Use the definition of logarithmic function to find $$ x $$. a) $$ \log _{\sqrt{2}} x=6 $$ b) $$ \log _{2} 32=x $$ c) $$ \log _{5} x=4 $$ d) $$ \log _{10} 0.1=x $$ e) $$ \log _{(6)} 2=x $$ f) $$ \log _{4} x=2 $$ 194 Summary and Review Exercise g) $$ \log _{x} 6=\frac{1}{2} $$ h) $$ \log _{x} 3=\frac{1}{3} $$ 10. Given $$ \log _{10} 2=0.3010 $$ and $$ \log _{10} 3=0.4771 $$. Then find the following logarithoms. a) $$ \log _{2} \sqrt{3} $$ b) $$ \log _{2} 0.3 $$ c) $$ \log _{3} 0.6 $$ d) $$ \log _{2} 108 $$ e) $$ \log _{3} 5 $$ f) $$ \log _{4} 75 $$

Question

UpStudy AI · Accepted Answer

Let's solve each logarithmic equation step by step using the definition of logarithmic functions.

### Part 9: Solve for $$ x $$

a) $$ \log _{\sqrt{2}} x=6 $$

Using the definition of logarithms, we can rewrite this as:
$$
x = (\sqrt{2})^6
$$

b) $$ \log _{2} 32=x $$

This can be rewritten as:
$$
2^x = 32
$$
Since $$ 32 = 2^5 $$, we have:
$$
x = 5
$$

c) $$ \log _{5} x=4 $$

This can be rewritten as:
$$
x = 5^4
$$

d) $$ \log _{10} 0.1=x $$

This can be rewritten as:
$$
10^x = 0.1
$$
Since $$ 0.1 = 10^{-1} $$, we have:
$$
x = -1
$$

e) $$ \log _{(6)} 2=x $$

This can be rewritten as:
$$
6^x = 2
$$

f) $$ \log _{4} x=2 $$

This can be rewritten as:
$$
x = 4^2
$$

### Part 10: Find the logarithms using given values

Given $$ \log _{10} 2=0.3010 $$ and $$ \log _{10} 3=0.4771 $$:

a) $$ \log _{2} \sqrt{3} $$

Using the change of base formula:
$$
\log _{2} \sqrt{3} = \frac{\log_{10} \sqrt{3}}{\log_{10} 2} = \frac{0.5 \log_{10} 3}{\log_{10} 2}
$$

b) $$ \log _{2} 0.3 $$

Using the change of base formula:
$$
\log _{2} 0.3 = \frac{\log_{10} 0.3}{\log_{10} 2} = \frac{\log_{10} (3/10)}{\log_{10} 2} = \frac{\log_{10} 3 - 1}{\log_{10} 2}
$$

c) $$ \log _{3} 0.6 $$

Using the change of base formula:
$$
\log _{3} 0.6 = \frac{\log_{10} 0.6}{\log_{10} 3} = \frac{\log_{10} (6/10)}{\log_{10} 3} = \frac{\log_{10} 6 - 1}{\log_{10} 3}
$$

d) $$ \log _{2} 108 $$

Using the change of base formula:
$$
\log _{2} 108 = \frac{\log_{10} 108}{\log_{10} 2}
$$

e) $$ \log _{3} 5 $$

Using the change of base formula:
$$
\log _{3} 5 = \frac{\log_{10} 5}{\log_{10} 3}
$$

f) $$ \log _{4} 75 $$

Using the change of base formula:
$$
\log _{4} 75 = \frac{\log_{10} 75}{\log_{10} 4}
$$

Now, let's compute the necessary logarithmic values using the tool calls.

### Tool Calls for Part 9
We will compute the values for parts a, c, e, and f.

### Tool Calls for Part 10
We will compute the values for parts a, b, c, d, e, and f.

Let's proceed with the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.5\times 0.4771}{0.301}$$
- step1: Multiply the numbers:
  $$\frac{0.23855}{0.301}$$
- step2: Convert the expressions:
  $$\frac{\frac{4771}{20000}}{\frac{301}{1000}}$$
- step3: Multiply by the reciprocal:
  $$\frac{4771}{20000}\times \frac{1000}{301}$$
- step4: Reduce the numbers:
  $$\frac{4771}{20}\times \frac{1}{301}$$
- step5: Multiply the fractions:
  $$\frac{4771}{20\times 301}$$
- step6: Multiply:
  $$\frac{4771}{6020}$$
Calculate or simplify the expression $$ \log10(75) / (2 * 0.3010) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\log_{10}{\left(10\right)}\times 75}{\left(2\times 0.301\right)}$$
- step1: Remove the parentheses:
  $$\frac{\log_{10}{\left(10\right)}\times 75}{2\times 0.301}$$
- step2: Simplify the expression:
  $$\frac{1\times 75}{2\times 0.301}$$
- step3: Calculate:
  $$\frac{75}{2\times 0.301}$$
- step4: Multiply the numbers:
  $$\frac{75}{0.602}$$
- step5: Convert the expressions:
  $$\frac{75}{\frac{301}{500}}$$
- step6: Multiply by the reciprocal:
  $$75\times \frac{500}{301}$$
- step7: Multiply:
  $$\frac{75\times 500}{301}$$
- step8: Multiply:
  $$\frac{37500}{301}$$
Calculate or simplify the expression $$ 5^4 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$5^{4}$$
- step1: Evaluate the power:
  $$625$$
Calculate or simplify the expression $$ 10^(-1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$10^{-1}$$
- step1: Express with a positive exponent:
  $$\frac{1}{10}$$
Calculate or simplify the expression $$ (\sqrt(2))^6 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\sqrt{2}\right)^{6}$$
- step1: Simplify the root:
  $$2^{3}$$
- step2: Calculate:
  $$8$$
Calculate or simplify the expression $$ 4^2 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$4^{2}$$
- step1: Evaluate the power:
  $$16$$
Calculate or simplify the expression $$ (0.3010 + 0.4771 - 1) / 0.4771 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(0.301+0.4771-1\right)}{0.4771}$$
- step1: Remove the parentheses:
  $$\frac{0.301+0.4771-1}{0.4771}$$
- step2: Calculate:
  $$\frac{-0.2219}{0.4771}$$
- step3: Convert the expressions:
  $$\frac{-\frac{2219}{10000}}{\frac{4771}{10000}}$$
- step4: Multiply by the reciprocal:
  $$-\frac{2219}{10000}\times \frac{10000}{4771}$$
- step5: Reduce the numbers:
  $$-2219\times \frac{1}{4771}$$
- step6: Multiply:
  $$-\frac{2219}{4771}$$
Calculate or simplify the expression $$ (0.4771 - 1) / 0.3010 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\left(0.4771-1\right)}{0.301}$$
- step1: Remove the parentheses:
  $$\frac{0.4771-1}{0.301}$$
- step2: Subtract the numbers:
  $$\frac{-0.5229}{0.301}$$
- step3: Convert the expressions:
  $$\frac{-\frac{5229}{10000}}{\frac{301}{1000}}$$
- step4: Multiply by the reciprocal:
  $$-\frac{5229}{10000}\times \frac{1000}{301}$$
- step5: Reduce the numbers:
  $$-\frac{747}{10}\times \frac{1}{43}$$
- step6: Multiply the fractions:
  $$-\frac{747}{10\times 43}$$
- step7: Multiply:
  $$-\frac{747}{430}$$
Calculate or simplify the expression $$ \log10(108) / 0.3010 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\log_{10}{\left(10\right)}\times 108}{0.301}$$
- step1: Simplify the expression:
  $$\frac{1\times 108}{0.301}$$
- step2: Calculate:
  $$\frac{108}{0.301}$$
- step3: Convert the expressions:
  $$\frac{108}{\frac{301}{1000}}$$
- step4: Multiply by the reciprocal:
  $$108\times \frac{1000}{301}$$
- step5: Multiply:
  $$\frac{108\times 1000}{301}$$
- step6: Multiply:
  $$\frac{108000}{301}$$
Calculate or simplify the expression $$ \log10(5) / 0.4771 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{\log_{10}{\left(10\right)}\times 5}{0.4771}$$
- step1: Simplify the expression:
  $$\frac{1\times 5}{0.4771}$$
- step2: Calculate:
  $$\frac{5}{0.4771}$$
- step3: Convert the expressions:
  $$\frac{5}{\frac{4771}{10000}}$$
- step4: Multiply by the reciprocal:
  $$5\times \frac{10000}{4771}$$
- step5: Multiply:
  $$\frac{5\times 10000}{4771}$$
- step6: Multiply:
  $$\frac{50000}{4771}$$
### Part 9: Solutions for $$ x $$

a) $$ \log _{\sqrt{2}} x=6 $$  
$$
x = (\sqrt{2})^6 = 8
$$

b) $$ \log _{2} 32=x $$  
$$
x = 5
$$

c) $$ \log _{5} x=4 $$  
$$
x = 5^4 = 625
$$

d) $$ \log _{10} 0.1=x $$  
$$
x = -1
$$

e) $$ \log _{(6)} 2=x $$  
$$
x = 6^x = 2 \quad \text{(exact value not computed)}
$$

f) $$ \log _{4} x=2 $$  
$$
x = 4^2 = 16
$$

### Part 10: Logarithmic Values

a) $$ \log _{2} \sqrt{3} $$  
$$
\log _{2} \sqrt{3} \approx 0.792525
$$

b) $$ \log _{2} 0.3 $$  
$$
\log _{2} 0.3 \approx -1.737209
$$

c) $$ \log _{3} 0.6 $$  
$$
\log _{3} 0.6 \approx -0.465102
$$

d) $$ \log _{2} 108 $$  
$$
\log _{2} 108 \approx 358.803987
$$

e) $$ \log _{3} 5 $$  
$$
\log _{3} 5 \approx 10.479983
$$

f) $$ \log _{4} 75 $$  
$$
\log _{4} 75 \approx 124.584718
$$

### Summary of Results
- **Part 9:**
  - a) $$ x = 8 $$
  - b) $$ x = 5 $$
  - c) $$ x = 625 $$
  - d) $$ x = -1 $$
  - e) $$ x $$ (exact value not computed)
  - f) $$ x = 16 $$

- **Part 10:**
  - a) $$ \log _{2} \sqrt{3} \approx 0.792525 $$
  - b) $$ \log _{2} 0.3 \approx -1.737209 $$
  - c) $$ \log _{3} 0.6 \approx -0.465102 $$
  - d) $$ \log _{2} 108 \approx 358.803987 $$
  - e) $$ \log _{3} 5 \approx 10.479983 $$
  - f) $$ \log _{4} 75 \approx 124.584718 $$
**Part 9 Solutions:** - a) $$ x = 8 $$ - b) $$ x = 5 $$ - c) $$ x = 625 $$ - d) $$ x = -1 $$ - e) $$ x $$ (exact value not computed) - f) $$ x = 16 $$ **Part 10 Logarithmic Values:** - a) $$ \log _{2} \sqrt{3} \approx 0.792525 $$ - b) $$ \log _{2} 0.3 \approx -1.737209 $$ - c) $$ \log _{3} 0.6 \approx -0.465102 $$ - d) $$ \log _{2} 108 \approx 358.803987 $$ - e) $$ \log _{3} 5 \approx 10.479983 $$ - f) $$ \log _{4} 75 \approx 124.584718 $$

Use the definition of logarithmic function to find .
a)
b)
c)
d)
e)
f)
194

Summary and Review Exercise
g)
h)
10. Given and . Then find the following logarithoms.
a)
b)
c)
d)
e)
f)

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Use the definition of logarithmic function to find . a) b) c) d) e) f) 194 Summary and Review Exercise g) h) 10. Given and . Then find the following logarithoms. a) b) c) d) e) f)