Given that , find the value of.
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Given that , find the value of.
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In this question,Ā andĀ are positive constants.
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The function is defined by.
Sketch the graph ofĀ and hence sketch the graph ofĀ on the axes below.
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Solve the equationĀ
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WriteĀ as a single logarithm.
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Using the substitutionĀ , or otherwise, solveĀ .
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For variablesĀ and , plottingĀ against gives a straight-line graph passing through the points and .
Show thatĀ =Ā whereĀ andĀ are integers to be found.
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VariablesĀ and are such that, when is plotted against , a straight line graph passing through the points (6,7) and (10,9) is obtained. Find as a function of .
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Given that , find the value of .
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VariablesĀ andĀ are connected by the relationship , whereĀ andĀ are constants.
Transform the relationshipĀ to straight line form.
When is plotted against a straight line graph passing through the points (0, 0.5) and (3.2, 1.7) is obtained.
Find the value ofĀ and of .
Find the value ofĀ when .
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The population , in millions, of a country is given by , whereĀ is the number of years after January 2000 andĀ andĀ are constants. In January 2010 the population was 40 million and had increased to 45 million by January 2013.
Show thatĀ 1.04 to 2 decimal places and findĀ to the nearest integer.
Find the population in January 2020, giving your answer to the nearest million.
In January of which year will the population be over 100 million for the first time?
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The number, , of bacteria in a sample is given byĀ , whereĀ andĀ are constants andĀ is time in weeks. Initially there are 500 bacteria which increase to 600 after 1 week.
Find the value ofĀ and of .
Find the number of bacteria present after 2 weeks.
Find the first week in which the number of bacteria is greater than 1 000 000.
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Find the exact solution ofĀ .
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Solve the equationĀ .
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Solve the equation .
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Write the expressionĀ in the form , whereĀ andĀ are integers.
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Find the exact solution of .
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Solve the simultaneous equations
,
Ā ,
givingĀ andĀ in exact simplified form.
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.
Solve this equation for , giving your answers in terms of .
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Solve the simultaneous equations.
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It is known thatĀ , whereĀ andĀ are constants. WhenĀ is plotted againstĀ , a straight line passing through the points (3.63, 5.25) and (4.83, 6.88) is obtained.
Find the value ofĀ and of .
Using your values ofĀ and , find the value ofĀ whenĀ ,
Find the positive value ofĀ when .
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Show that .
Find the roots of .
Give a reason why only one root is a valid solution of the logarithmic equations. Find the value ofĀ corresponding to this root.
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