7. Logarithms

Answer: $4 x^{3}$

The power rule turns $3\log_{3}(x)$ into $\log_{3}(x^{3})$, and the product rule combines the two into $\log_{3}(4 x^{3})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{3} + 4$
• Multiplied by the power instead of raising to it → $12 x$

Answer: $3$

Equal logarithms with the same base have equal arguments, so $x + 4 = 7$ and $x = 3$. Check the domain: $x + 4 > 0$ holds.

Common mistakes, and the answer each one gives:

• Subtracted the shift from the wrong side → $11$
• Treated the log as a multiplier and divided → $\frac{7}{4}$

Answer: $- \frac{1}{3}$

Write the right-hand side as a power of $3$: $243 = 3^{5}$. With the same base on both sides the exponents must be equal, so $3 x + 6 = 5$ and $x = - \frac{1}{3}$.

Common mistakes, and the answer each one gives:

• Set the exponent equal to 243 instead of to 5 → $79$
• Forgot to move the constant in the exponent across → $\frac{5}{3}$

Answer: $2$

Write both numbers as powers of $3$: $9 = 3^{2}$ and $81 = 3^{4}$. Then $\log_{9}81 = \dfrac{4}{2} = 2$, because the base has to be raised to that power to reach the argument.

Common mistakes, and the answer each one gives:

• Divided the two numbers → $9$
• Turned the fraction of exponents upside down → $\frac{1}{2}$

Answer: $5 x^{2}$

The power rule turns $2\log_{5}(x)$ into $\log_{5}(x^{2})$, and the product rule combines the two into $\log_{5}(5 x^{2})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{2} + 5$
• Multiplied by the power instead of raising to it → $10 x$

Answer: $8$

Equal logarithms with the same base have equal arguments, so $x + 1 = 9$ and $x = 8$. Check the domain: $x + 1 > 0$ holds.

Common mistakes, and the answer each one gives:

• Subtracted the shift from the wrong side → $10$
• Treated the log as a multiplier and divided → $9$

Answer: $\frac{3}{4}$

Write the right-hand side as a power of $2$: $4 = 2^{2}$. With the same base on both sides the exponents must be equal, so $4 x - 1 = 2$ and $x = \frac{3}{4}$.

Common mistakes, and the answer each one gives:

• Set the exponent equal to 4 instead of to 2 → $\frac{5}{4}$
• Forgot to move the constant in the exponent across → $\frac{1}{2}$

Answer: $\frac{3}{2}$

Write both numbers as powers of $5$: $25 = 5^{2}$ and $125 = 5^{3}$. Then $\log_{25}125 = \dfrac{3}{2} = \frac{3}{2}$, because the base has to be raised to that power to reach the argument.

Common mistakes, and the answer each one gives:

• Divided the two numbers → $5$
• Turned the fraction of exponents upside down → $\frac{2}{3}$

Answer: $6 x^{4}$

The power rule turns $4\log_{7}(x)$ into $\log_{7}(x^{4})$, and the product rule combines the two into $\log_{7}(6 x^{4})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{4} + 6$
• Multiplied by the power instead of raising to it → $24 x$

Answer: $7$

Equal logarithms with the same base have equal arguments, so $x + 7 = 14$ and $x = 7$. Check the domain: $x + 7 > 0$ holds.

Common mistakes, and the answer each one gives:

• Subtracted the shift from the wrong side → $21$
• Treated the log as a multiplier and divided → $2$

Answer: $4$

Write the right-hand side as a power of $2$: $32 = 2^{5}$. With the same base on both sides the exponents must be equal, so $2 x - 3 = 5$ and $x = 4$.

Common mistakes, and the answer each one gives:

• Set the exponent equal to 32 instead of to 5 → $\frac{35}{2}$
• Forgot to move the constant in the exponent across → $\frac{5}{2}$

Answer: $\frac{5}{2}$

Write both numbers as powers of $2$: $4 = 2^{2}$ and $32 = 2^{5}$. Then $\log_{4}32 = \dfrac{5}{2} = \frac{5}{2}$, because the base has to be raised to that power to reach the argument.

Common mistakes, and the answer each one gives:

• Divided the two numbers → $8$
• Turned the fraction of exponents upside down → $\frac{2}{5}$

Answer: $6 x^{5}$

The power rule turns $5\log_{7}(x)$ into $\log_{7}(x^{5})$, and the product rule combines the two into $\log_{7}(6 x^{5})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{5} + 6$
• Multiplied by the power instead of raising to it → $30 x$

Answer: $5$

Equal logarithms with the same base have equal arguments, so $x + 8 = 13$ and $x = 5$. Check the domain: $x + 8 > 0$ holds.

Common mistakes, and the answer each one gives:

• Subtracted the shift from the wrong side → $21$
• Treated the log as a multiplier and divided → $\frac{13}{8}$

Answer: $0$

Write the right-hand side as a power of $5$: $25 = 5^{2}$. With the same base on both sides the exponents must be equal, so $2 x + 2 = 2$ and $x = 0$.

Common mistakes, and the answer each one gives:

• Set the exponent equal to 25 instead of to 2 → $\frac{23}{2}$
• Forgot to move the constant in the exponent across → $1$

Answer: $\frac{5}{3}$

Write both numbers as powers of $2$: $8 = 2^{3}$ and $32 = 2^{5}$. Then $\log_{8}32 = \dfrac{5}{3} = \frac{5}{3}$, because the base has to be raised to that power to reach the argument.

Common mistakes, and the answer each one gives:

• Divided the two numbers → $4$
• Turned the fraction of exponents upside down → $\frac{3}{5}$

Answer: $2 x^{6}$

The power rule turns $6\log_{3}(x)$ into $\log_{3}(x^{6})$, and the product rule combines the two into $\log_{3}(2 x^{6})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{6} + 2$
• Multiplied by the power instead of raising to it → $12 x$

Answer: $4$

Equal logarithms with the same base have equal arguments, so $x + 4 = 8$ and $x = 4$. Check the domain: $x + 4 > 0$ holds.

Common mistakes, and the answer each one gives:

• Subtracted the shift from the wrong side → $12$
• Treated the log as a multiplier and divided → $2$

Answer: $4$

Write the right-hand side as a power of $3$: $81 = 3^{4}$. With the same base on both sides the exponents must be equal, so $2 x - 4 = 4$ and $x = 4$.

Common mistakes, and the answer each one gives:

• Set the exponent equal to 81 instead of to 4 → $\frac{85}{2}$
• Forgot to move the constant in the exponent across → $2$

Answer: $3$

Write both numbers as powers of $5$: $25 = 5^{2}$ and $15625 = 5^{6}$. Then $\log_{25}15625 = \dfrac{6}{2} = 3$, because the base has to be raised to that power to reach the argument.

Common mistakes, and the answer each one gives:

• Divided the two numbers → $625$
• Turned the fraction of exponents upside down → $\frac{1}{3}$

Answer: $7 x^{4}$

The power rule turns $4\log_{3}(x)$ into $\log_{3}(x^{4})$, and the product rule combines the two into $\log_{3}(7 x^{4})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{4} + 7$
• Multiplied by the power instead of raising to it → $28 x$

Answer: $11$

Equal logarithms with the same base have equal arguments, so $x + 2 = 13$ and $x = 11$. Check the domain: $x + 2 > 0$ holds.

Common mistakes, and the answer each one gives:

• Subtracted the shift from the wrong side → $15$
• Treated the log as a multiplier and divided → $\frac{13}{2}$

Answer: $\frac{3}{2}$

Write the right-hand side as a power of $2$: $32 = 2^{5}$. With the same base on both sides the exponents must be equal, so $4 x - 1 = 5$ and $x = \frac{3}{2}$.

Common mistakes, and the answer each one gives:

• Set the exponent equal to 32 instead of to 5 → $\frac{33}{4}$
• Forgot to move the constant in the exponent across → $\frac{5}{4}$

Answer: $\frac{4}{3}$

Write both numbers as powers of $5$: $125 = 5^{3}$ and $625 = 5^{4}$. Then $\log_{125}625 = \dfrac{4}{3} = \frac{4}{3}$, because the base has to be raised to that power to reach the argument.

Common mistakes, and the answer each one gives:

• Divided the two numbers → $5$
• Turned the fraction of exponents upside down → $\frac{3}{4}$

Answer: $6 x^{2}$

The power rule turns $2\log_{10}(x)$ into $\log_{10}(x^{2})$, and the product rule combines the two into $\log_{10}(6 x^{2})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{2} + 6$
• Multiplied by the power instead of raising to it → $12 x$

Answer: $10$

Equal logarithms with the same base have equal arguments, so $x + 4 = 14$ and $x = 10$. Check the domain: $x + 4 > 0$ holds.

Common mistakes, and the answer each one gives:

• Subtracted the shift from the wrong side → $18$
• Treated the log as a multiplier and divided → $\frac{7}{2}$

Answer: $\frac{4}{3}$

Write the right-hand side as a power of $2$: $8 = 2^{3}$. With the same base on both sides the exponents must be equal, so $3 x - 1 = 3$ and $x = \frac{4}{3}$.

Common mistakes, and the answer each one gives:

• Set the exponent equal to 8 instead of to 3 → $3$
• Forgot to move the constant in the exponent across → $1$

Answer: $\frac{4}{3}$

Write both numbers as powers of $3$: $27 = 3^{3}$ and $81 = 3^{4}$. Then $\log_{27}81 = \dfrac{4}{3} = \frac{4}{3}$, because the base has to be raised to that power to reach the argument.

Common mistakes, and the answer each one gives:

• Divided the two numbers → $3$
• Turned the fraction of exponents upside down → $\frac{3}{4}$

Answer: $7 x^{6}$

The power rule turns $6\log_{6}(x)$ into $\log_{6}(x^{6})$, and the product rule combines the two into $\log_{6}(7 x^{6})$.

Common mistakes, and the answer each one gives:

• Added the arguments instead of multiplying them → $x^{6} + 7$
• Multiplied by the power instead of raising to it → $42 x$

Answer: $6$

Equal logarithms with the same base have equal arguments, so $x + 4 = 10$ and $x = 6$. Check the domain: $x + 4 > 0$ holds.

Common mistakes, and the answer each one gives:

• Subtracted the shift from the wrong side → $14$
• Treated the log as a multiplier and divided → $\frac{5}{2}$