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High School Math : Mathematical Relationships and Basic Graphs

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Example Questions

Example Question #1 : Simplifying Logarithms

Which of the following expressions is equivalent to $\displaystyle log(2)^3$?

Possible Answers:

log(5) $\displaystyle log(5)$

log(6) $\displaystyle log(6)$

$\displaystyle 3*log(2)$

log(8) $\displaystyle log(8)$

$\displaystyle log(2) + 3$

Correct answer:

$\displaystyle 3*log(2)$

Explanation:

According to the rule for exponents of logarithms,$\displaystyle log(a)^b = b*log(a)$. As a direct application of this,$\displaystyle log(2)^3 = 3*log(2)$.

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Example Question #4 : Simplifying Logarithms

Simplify the expression below.

$log(2 \cdot 2 \cdot 2)$ $\displaystyle log(2 \cdot 2 \cdot 2)$

Possible Answers:

8log(2) $\displaystyle 8log(2)$

4log(2) $\displaystyle 4log(2)$

log(6) $\displaystyle log(6)$

3log(2) $\displaystyle 3log(2)$

log(5) $\displaystyle log(5)$

Correct answer:

3log(2) $\displaystyle 3log(2)$

Explanation:

Based on the definition of exponents, $log(2 \cdot 2 \cdot 2) = log (2^{3})$ $\displaystyle log(2 \cdot 2 \cdot 2) = log (2^{3})$.

Then, we use the following rule of logarithms:

$log(m^{n}) = n\cdot log(m)$ $\displaystyle log(m^{n}) = n\cdot log(m)$

Thus, $log(2^{3}) = 3log(2)$ $\displaystyle log(2^{3}) = 3log(2)$.

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Example Question #1 : Solving Logarithmic Equations

Solve the equation.

$3^{3x-7}=81^{12-3x}$ $\displaystyle 3^{3x-7}=81^{12-3x}$

Possible Answers:

x=11 $\displaystyle x=11$

$x=\frac{11}{3}$ $\displaystyle x=\frac{11}{3}$

x=33 $\displaystyle x=33$

$x=\frac{22}{3}$ $\displaystyle x=\frac{22}{3}$

x=9 $\displaystyle x=9$

Correct answer:

$x=\frac{11}{3}$ $\displaystyle x=\frac{11}{3}$

Explanation:

Change 81 to 3^4 $\displaystyle 3^4$ so that both sides have the same base. Once you have the same base, apply log to both sides so that you can set the exponential expressions equal to each other ($\displaystyle 3x-7 = 48-12x$). Thus, $x=\frac{11}{3}$ $\displaystyle x=\frac{11}{3}$.

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Example Question #1 : Solving Logarithmic Equations

Solve the equation.

$36^{10x+2}=(\frac{1}{6})^{13-x}$ $\displaystyle 36^{10x+2}=(\frac{1}{6})^{13-x}$

Possible Answers:

x=19 $\displaystyle x=19$

x=17 $\displaystyle x=17$

$x=-\frac{17}{19}$ $\displaystyle x=-\frac{17}{19}$

$x=\frac{17}{19}$ $\displaystyle x=\frac{17}{19}$

$x=\frac{11}{19}$ $\displaystyle x=\frac{11}{19}$

Correct answer:

$x=-\frac{17}{19}$ $\displaystyle x=-\frac{17}{19}$

Explanation:

Change the left side to 6^2 $\displaystyle 6^2$ and the right side to $6^{-1}$ $\displaystyle 6^{-1}$ so that both sides have the same base. Apply log to both sides and then set the exponential expressions equal to each other ($\displaystyle 20x+4=-13+x$). $x=-\frac{17}{19}$ $\displaystyle x=-\frac{17}{19}$.

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Example Question #1 : Solving And Graphing Logarithmic Equations

Solve the equation.

$8^{x-1}=(\frac{1}{16})^{4x-3}$ $\displaystyle 8^{x-1}=(\frac{1}{16})^{4x-3}$

Possible Answers:

x=30 $\displaystyle x=30$

$x=\frac{11}{19}$ $\displaystyle x=\frac{11}{19}$

x=13 $\displaystyle x=13$

x=15 $\displaystyle x=15$

$x=\frac{15}{19}$ $\displaystyle x=\frac{15}{19}$

Correct answer:

$x=\frac{15}{19}$ $\displaystyle x=\frac{15}{19}$

Explanation:

Change the left side to 2^3 $\displaystyle 2^3$ and the right side to $2^{-4}$ $\displaystyle 2^{-4}$ so that both sides have the same base. Apply log and then set the exponential expressions equal to each other ($\displaystyle 3x-3=-16x+12$). Thus, $x=\frac{15}{19}$ $\displaystyle x=\frac{15}{19}$.

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Example Question #151 : Mathematical Relationships And Basic Graphs

Solve the equation.

$25^{20x+4}=(\frac{1}{125})^{4-3x}$ $\displaystyle 25^{20x+4}=(\frac{1}{125})^{4-3x}$

Possible Answers:

$x=-\frac{11}{31}$ $\displaystyle x=-\frac{11}{31}$

$x=\frac{9}{40}$ $\displaystyle x=\frac{9}{40}$

$x=-\frac{20}{31}$ $\displaystyle x=-\frac{20}{31}$

x=18 $\displaystyle x=18$

$x=\frac{20}{31}$ $\displaystyle x=\frac{20}{31}$

Correct answer:

$x=-\frac{20}{31}$ $\displaystyle x=-\frac{20}{31}$

Explanation:

Change the left side to 5^2 $\displaystyle 5^2$ and the right side to $5^{-3}$ $\displaystyle 5^{-3}$ so that both sides have the same base. Apply log and then set the exponential expressions equal to each other ($\displaystyle 40x+8=-12+9x$). Thus, $x=-\frac{20}{31}$ $\displaystyle x=-\frac{20}{31}$

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Example Question #1 : Solving Logarithmic Equations

Solve the equation.

$27^{4x-1}=(\frac{1}{3})^{3x+8}$ $\displaystyle 27^{4x-1}=(\frac{1}{3})^{3x+8}$

Possible Answers:

x=1 $\displaystyle x=1$

$x=-\frac{1}{3}$ $\displaystyle x=-\frac{1}{3}$

x=-1 $\displaystyle x=-1$

$x=\frac{1}{3}$ $\displaystyle x=\frac{1}{3}$

x=3 $\displaystyle x=3$

Correct answer:

$x=-\frac{1}{3}$ $\displaystyle x=-\frac{1}{3}$

Explanation:

Change the left side to 3^3 $\displaystyle 3^3$ and the right side to $3^{-1}$ $\displaystyle 3^{-1}$ so that both sides have the same base. Apply log to both sides and then set the exponential expressions equal to each other ($\displaystyle 12x-3=-3x-8$). Thus, $x=-\frac{1}{3}$ $\displaystyle x=-\frac{1}{3}$.

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Example Question #151 : Mathematical Relationships And Basic Graphs

Solve for $\displaystyle x$.

$17 + x = \log_{10}10000$ $\displaystyle 17 + x = \log_{10}10000$

Possible Answers:

$\displaystyle 10^4 - 17$

$\displaystyle -13$

983 $\displaystyle 983$

$\displaystyle 4$

10^1^7 $\displaystyle 10^1^7$

Correct answer:

$\displaystyle -13$

Explanation:

$log_{10}10000$ $\displaystyle log_{10}10000$ can be simplified to $\displaystyle 4$ since $\displaystyle 10^4 = 10000$. This gives the equation:

$\displaystyle 17 + x = 4$

Subtracting $\displaystyle 17$ from both sides of the equation gives the value for $\displaystyle x$.

$(17 + x) - 17 = (4) - 17 \rightarrow x = -13$ $\displaystyle (17 + x) - 17 = (4) - 17 \rightarrow x = -13$

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Example Question #152 : Mathematical Relationships And Basic Graphs

Solve the equation.

$5^{3x-9}=25^{^{x+5}$ $\displaystyle 5^{3x-9}=25^{^{x+5}$

Possible Answers:

x=10 $\displaystyle x=10$

x=11 $\displaystyle x=11$

x=5 $\displaystyle x=5$

x=19 $\displaystyle x=19$

x=21 $\displaystyle x=21$

Correct answer:

x=19 $\displaystyle x=19$

Explanation:

First, change 25 to 5^2 $\displaystyle 5^2$ so that both sides have the same base. Once they have the same base, you can apply log to both sides so that you can set their exponents equal to each other, which yields $\displaystyle 3x-9=2x+10$.

x=19 $\displaystyle x=19$

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Example Question #8 : Solving And Graphing Logarithmic Equations

Solve the equation.

$7^{3x+4}=49^{x+1}$ $\displaystyle 7^{3x+4}=49^{x+1}$

Possible Answers:

x=-4 $\displaystyle x=-4$

x=-2 $\displaystyle x=-2$

x=-1 $\displaystyle x=-1$

x=4 $\displaystyle x=4$

x=2 $\displaystyle x=2$

Correct answer:

x=-2 $\displaystyle x=-2$

Explanation:

Change 49 to 7^2 $\displaystyle 7^2$ so that both sides have the same base so that you can apply log. Then, you can set the exponential expressions equal to each other $\displaystyle 3x+4=2x+2$.

Thus, x=-2 $\displaystyle x=-2$

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High School Math : Mathematical Relationships and Basic Graphs

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Simplifying Logarithms

Example Question #4 : Simplifying Logarithms

Example Question #1 : Solving Logarithmic Equations

Example Question #1 : Solving Logarithmic Equations

Example Question #1 : Solving And Graphing Logarithmic Equations

Example Question #151 : Mathematical Relationships And Basic Graphs

Example Question #1 : Solving Logarithmic Equations

Example Question #151 : Mathematical Relationships And Basic Graphs

Example Question #152 : Mathematical Relationships And Basic Graphs

Example Question #8 : Solving And Graphing Logarithmic Equations

All High School Math Resources

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