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Question 1 of 4
Derive the Change of Base formula from the general equation
x=logaNx=logaN
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Logarithmic Form
x=logaNx=logaN
Transform the general logarithmic equation to exponent form
xx |
== |
logaNlogaN |
NN |
== |
axax |
Insert logarithms of the same base to both sides, then solve for xx
NN |
== |
axax |
logbNlogbN |
== |
logbaxlogbax |
logbNlogbN |
== |
logbaxlogbax |
logbNlogbN |
== |
xlogbaxlogba |
logbxp=plogbxlogbxp=plogbx |
logbNlogbN÷logba÷logba |
== |
xlogbaxlogba÷logba÷logba |
Divide both sides by logbalogba |
|
logbNlogbalogbNlogba |
== |
xx |
|
xx |
== |
logbNlogbalogbNlogba |
Also, remember that, x=logaNx=logaN
Hence, the Change of Base formula is: logaN=logbNlogbalogaN=logbNlogba
logaN=logbNlogbalogaN=logbNlogba
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Question 2 of 4
Evaluate using Change of Base
log29log29
Round answer to 55 decimal places
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Logarithmic Form
x=logaNx=logaN
Use the change of base formula, then use the calculator to solve the logarithm
log29log29 |
== |
log109log102log109log102 |
Calculators use 1010 as base for the log function |
|
|
== |
3.169933.16993 |
Compute using the calculator |
log29=3.16993log29=3.16993
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Question 3 of 4
Solve for xx using Change of Base
5x=115x=11
Round answer to 44 decimal places
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Logarithmic Form
x=logaNx=logaN
Transform the given exponential equation to logarithmic form
5x5x |
== |
1111 |
xx |
== |
log511log511 |
Use the change of base formula, then use the calculator to solve the logarithm
xx |
== |
log511log511 |
|
x |
= |
log1011log105 |
Calculators use 10 as base for the log function |
|
x |
= |
1.4899 |
Compute using the calculator |
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Question 4 of 4
Solve for x using Change of Base
2x=0.062
Round answer to 4 decimal places
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Transform the given exponential equation to logarithmic form
Use the change of base formula, then use the calculator to solve the logarithm
x |
= |
log20.062 |
|
x |
= |
log100.062log102 |
Calculators use 10 as base for the log function |
|
x |
= |
-4.0116 |
Compute using the calculator |