smc-4243-70-Solve exponentials

Logarithm, SMB-024

Solve `log_9 27=x` for `x`. (2 marks)

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`3/2`

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`log_9 27`	`=x`
`9^x`	`=27\ \ text{(by log definition)}`
`log_10 9^x`	`=log_10 27`
`x log_10 9`	`=log_10 27`
`x`	`=(log_10 27)/(log_10 9)`
	`=3/2`

Logarithm, SMB-023

Solve `log_16 2=x` for `x`. (2 marks)

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`0.25`

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`log_16 2`	`=x`
`16^x`	`=2`
`log_10 16^x`	`=log_10 2`
`x log_10 16`	`=log_10 2`
`x`	`=(log_10 2)/(log_10 16)`
	`=0.25`

Logarithm, SMB-022

Solve `4^(x-1)=84` for `x`, giving your answer correct to 1 decimal place. (2 marks)

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`4.2`

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`4^(x-1)`	`=84`
`log_10 4^(x-1)`	`=log_10 84`
`(x-1)log_10 4`	`=log_10 84`
`x-1`	`=(log_10 84)/(log_10 4)`
`x`	`=(log_10 84)/(log_10 4)+1`
	`=4.196…`
	`=4.2\ text{(to 1 d.p.)}`

Logarithm, SMB-021

Solve `3^a=28` for `a`, giving your answer correct to 2 decimal places. (2 marks)

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`3.03`

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`3^a`	`=28`
`log_10 3^a`	`=log_10 28`
`a xx log_10 3`	`=log_10 28`
`a`	`=(log_10 28)/(log_10 3)`
	`=3.033…`
	`=3.03`

Logarithm, SMB-020

Solve `4^(x+1)=32` for `x`. (2 marks)

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`3/2`

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`4^(x+1)`	`=32`
`log_10 4^(x+1)`	`=log_10 32`
`(x+1) log_10 4`	`=log_10 32`
`x+1`	`=(log_10 32)/(log_10 4)`
`x`	`=5/2-1`
	`=3/2`

Logarithm, SMB-019

Solve `2^t=16` . (2 marks)

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`4`

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`2^t`	`=16`
`log_10 2^t`	`=log_10 16`
`t xx log_10 2`	`=log_10 16`
`t`	`=(log_10 16)/(log_10 2)`
	`=4`