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Question 1 of 5
Solve for x using Change of Base
3x+1=8
Round answer to 4 decimal places
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Logarithmic Form
x=logaN
Transform the given exponential equation to logarithmic form
3x+1 |
= |
8 |
x+1 |
= |
log38 |
Use the change of base formula, then use the calculator to solve the logarithm
x+1 |
= |
log38 |
|
x+1 |
= |
log108log103 |
Calculators use 10 as base for the log function |
|
x+1 |
= |
1.8928 |
Compute using the calculator |
x+1 −1 |
= |
1.8928 −1 |
Subtract 1 from both sides |
x |
= |
0.8928 |
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Question 2 of 5
Solve for x
2x=5x−1
Round answer to 4 decimal places
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Insert logarithms of the same base to both sides, then solve for x
2x |
= |
5x−1 |
log102x |
= |
log105x−1 |
Calculators use base 10 for log function |
xlog102 |
= |
(x−1)log105 |
logbxp=plogbx |
xlog102 |
= |
xlog105−log105 |
Distribute |
xlog102 +log105 |
= |
xlog105−log105 +log105 |
Add log105 to both sides |
xlog102+log105 |
= |
xlog105 |
xlog102+log105 −xlog102 |
= |
xlog105 −xlog102 |
Subtract xlog102 from both sides |
log105 |
= |
x(log105−log102) |
Factorize |
|
log105log105−log102 |
= |
x(log105−log102)log105−log102 |
Divide both sides by log105−log102 |
|
log105log105−log102 |
= |
x |
|
x |
= |
1.7565 |
Compute using the calculator |
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Question 3 of 5
Solve for x
103x−2=54x
Round answer to 3 decimal places
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Insert logarithms of the same base to both sides, then solve for x
103x−2 |
= |
54x |
log10103x−2 |
= |
log1054x |
Calculators use base 10 for log function |
(3x−2)log1010 |
= |
(4x)log105 |
logbxp=plogbx |
(3x−2)(1) |
= |
(4x)log105 |
logbb=1 |
3x−2 +2 |
= |
4xlog105 +2 |
Add 2 to both sides |
3x |
= |
4xlog105+2 |
3x −4xlog105 |
= |
4xlog105+2 −4xlog105 |
Subtract 4xlog105 from both sides |
x(3−4log105) |
= |
2 |
Factorize |
|
x(3−4log105)3−4log105 |
= |
23−4log105 |
Divide both sides by 3−4log105 |
|
x |
= |
23−4log105 |
|
x |
= |
9.798 |
Compute using the calculator |
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Question 4 of 5
Solve for x
6x=0.00025
Round answer to 3 decimal places
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Transform the decimal to exponential form
6x |
= |
0.00025 |
6x |
= |
25100000 |
0.00025=25100000 |
|
6x |
= |
14000 |
Simplify |
|
6x |
= |
4000−1 |
Reciprocate 14000 |
Insert logarithms of the same base to both sides, then solve for x
6x |
= |
4000−1 |
log106x |
= |
log104000−1 |
Calculators use 10 as base for the log function |
xlog106 |
= |
(−1)log104000 |
logbxp=plogbx |
|
xlog106log106 |
= |
(−1)log104000log106 |
Divide both sides by log106 |
|
x |
= |
(−1)log104000log106 |
|
x |
= |
−4.629 |
Compute using calculator |
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Question 5 of 5
Solve for x
(35)x=10−5
Round answer to 4 decimal places
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Insert logarithms of the same base to both sides, then solve for x
35x |
= |
10−5 |
|
log1035x |
= |
log1010−5 |
Calculators use 10 as base for the log function |
|
xlog1035 |
= |
(−5)log1010 |
logbxp=plogbx |
|
xlog1035 |
= |
(−5)(1) |
logbb=1 |
|
xlog1035log1035 |
= |
−5log1035 |
Divide both sides by log10(35) |
|
x |
= |
−5log10(35) |
|
x |
= |
−22.5379 |
Compute using the calculator |