// Numbas version: finer_feedback_settings {"name": "Nuala's copy of Logarithms: The definition", "feedback": {"showtotalmark": true, "advicethreshold": 0, "showanswerstate": true, "showactualmark": true, "allowrevealanswer": true, "enterreviewmodeimmediately": true, "showexpectedanswerswhen": "inreview", "showpartfeedbackmessageswhen": "always", "showactualmarkwhen": "always", "showtotalmarkwhen": "always", "showanswerstatewhen": "always", "showadvicewhen": "never"}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "allQuestions": true, "shuffleQuestions": false, "percentPass": 0, "duration": 0, "pickQuestions": 0, "navigation": {"onleave": {"action": "none", "message": ""}, "reverse": true, "allowregen": true, "showresultspage": "oncompletion", "preventleave": true, "browse": true, "showfrontpage": true}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "exam", "questions": [], "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": [{"name": "Logs: definition and concrete numbers", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "tags": ["logarithms", "Logarithms", "logs", "Logs"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "

The following should be completed without the use of a calculator.

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The definition of a logarithm says if $b$ and $a$ are positive and $b$ is not equal to 1, then $\\log_b(a)=c$ is equivalent to:

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The definition of a logarithm says if $b$ and $a$ are positive and $b$ is not equal to 1, then $\\log_b(a)=c$ is equivalent to $b^c=a$.

This means to determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

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$b^c=a$

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$b^a=c$

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$a^b=c$

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$a^c=b$

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$c^a=b$

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$c^b=a$

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The definition of a logarithm says that if $x$ and $z$ are positive and $x$ is not equal to 1, then $x^y=z$ is equivalent to:

The definition of a logarithm says if $b$ and $a$ are positive and $b$ is not equal to 1, then $\\log_b(a)=c$ is equivalent to $b^c=a$.

This means to determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

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$\\log_x (z)=y$

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$\\log_x (y)=z$

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$\\log_y (x)=z$

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$\\log_y (z)=x$

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$\\log_z (y)=x$

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$\\log_z (x)=y$

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Using the definition and your times tables (or index laws) determine the following:

$\\log_{\\var{zero[0]}}(\\var{zero[0]^zero[1]})$ = [[0]]

To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

To determine $\\log_{\\var{zero[0]}}(\\var{zero[0]^zero[1]})$, realise $\\var{zero[0]}^0=\\var{zero[0]^zero[1]}$ and so $\\log_{\\var{zero[0]}}(\\var{zero[0]^zero[1]})=0$.

Using the definition and your times tables (or index laws) determine the following:

$\\log_{\\var{one[0]}}(\\var{one[0]^one[1]})$ = [[0]]

To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

To determine $\\log_{\\var{one[0]}}(\\var{one[0]^one[1]})$, realise $\\var{one[0]}^1=\\var{one[0]}$ and so $\\log_{\\var{one[0]}}(\\var{one[0]^one[1]})=1$.

Using the definition and your times tables (or index laws) determine the following:

$\\log_{\\var{two[0]}}(\\var{two[0]^two[1]})$ = [[0]]

To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

To determine $\\log_{\\var{two[0]}}(\\var{two[0]^two[1]})$, realise $\\var{two[0]}^2=\\var{two[0]^2}$ and so $\\log_{\\var{two[0]}}(\\var{two[0]^two[1]})=2$.

Using the definition and your times tables (or index laws) determine the following:

$\\log_{\\var{small[0]}}(\\var{small[0]^small[1]})$ = [[0]]

To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

To determine $\\log_{\\var{small[0]}}(\\var{small[0]^small[1]})$, realise $\\var{small[0]}^\\var{small[1]}=\\var{small[0]^small[1]}$ and so $\\log_{\\var{small[0]}}(\\var{small[0]^small[1]})=\\var{small[1]}$.

Using the definition and your times tables (or index laws) determine the following:

$\\log_{\\var{tens[0]}}(\\var{tens[0]^tens[1]})$ = [[0]]

To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

To determine $\\log_{\\var{tens[0]}}(\\var{tens[0]^tens[1]})$, realise $\\var{tens[0]}^\\var{tens[1]}=\\var{tens[0]^tens[1]}$ and so $\\log_{\\var{tens[0]}}(\\var{tens[0]^tens[1]})=\\var{tens[1]}$.

Recall that $10^n$ is the same as a $1$ with $n$ zeros behind it.

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The following should be completed without the use of a calculator.

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Recall that we can convert negative indices to fractions

$\\var{num1}^{-1}$ = [[0]]

$\\var{num2}^{-2}$ = [[1]]

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When learning index laws you would have seen that
\\[x^{-n}=\\frac{1}{x^n}.\\]

Indices and logs are intimately related, ensure you revise your index laws.

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Using the definition and your times tables (or index laws) determine the following:

$\\log_{\\var{one[0]}}\\left(\\frac{1}{\\var{one[0]}}\\right)$ = [[0]]

To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

To determine $\\log_{\\var{one[0]}}\\left(\\frac{1}{\\var{one[0]}}\\right)$, realise $\\var{one[0]}^{-1}=\\frac{1}{\\var{one[0]}}$ and so $\\log_{\\var{one[0]}}\\left(\\frac{1}{\\var{one[0]}}\\right)=-1$.

Using the definition and your times tables (or index laws) determine the following:

$\\log_{\\var{two[0]}}\\left(\\frac{1}{\\var{two[0]^2}}\\right)$ = [[0]]

To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

To determine $\\log_{\\var{two[0]}}\\left(\\frac{1}{\\var{two[0]^2}}\\right)$, realise $\\var{two[0]}^{-2}=\\frac{1}{\\var{two[0]^2}}$ and so $\\log_{\\var{two[0]}}\\left(\\frac{1}{\\var{two[0]^2}}\\right)=-2$.

Using the definition and your times tables (or index laws) determine the following:

$\\log_{\\var{small[0]}}\\left(\\frac{1}{\\var{small[0]^small[1]}}\\right)$ = [[0]]

To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

To determine $\\log_{\\var{small[0]}}\\left(\\frac{1}{\\var{small[0]^small[1]}}\\right)$, realise $\\var{small[0]}^{-\\var{small[1]}}=\\frac{1}{\\var{small[0]^small[1]}}$ and so $\\log_{\\var{small[0]}}\\left(\\frac{1}{\\var{small[0]^small[1]}}\\right)=-\\var{small[1]}$.

Using the definition and your times tables (or index laws) determine the following:

$\\log_{\\var{tens[0]}}(\\var{1/tens[0]^tens[1]})$ = [[0]]

To determine the value of $\\log_b(a)$, you can think \"$b$ to the what equals $a$\", and that will be your answer.

To determine $\\log_{\\var{tens[0]}}(\\var{1/tens[0]^tens[1]})$, realise $\\var{tens[0]}^{-\\var{tens[1]}}=\\var{1/tens[0]^tens[1]}$ and so $\\log_{\\var{tens[0]}}(\\var{1/tens[0]^tens[1]})=-\\var{tens[1]}$.

Recall that $10^{-n}$ is the same as a decimal with zeros everywhere except a $1$ at the $n$th decimal place.