Grade 11 · Logarithms and domains · 396 of 600
Logarithms and domains · Domain boundary a=-2; Logarithm base b=4
Use x=1.08; the logarithm argument is exactly 3.08. State the domain, evaluate the logarithm, and recover the argument by exponentiation. Givens: Domain boundary a=-2; Logarithm base b=4.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
Let f(x)=log_4(x−(-2)). The base is 4>1. Use x=1.08; the logarithm argument is exactly 3.08. State the domain, evaluate the logarithm, and recover the argument by exponentiation.
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A starting point
Check x−a>0 first, then use ln(x−a)/ln(b). Exponentiating returns the argument before adding the shift.
Work through the reasoning
Step 1
Identify the model and target
Let f(x)=log_4(x−(-2)). The base is 4>1. Use x=1.08; the logarithm argument is exactly 3.08. The governing relation is y=log_b(x−a); bʸ=x−a; x>a. Check x−a>0 first, then use ln(x−a)/ln(b). Exponentiating returns the argument before adding the shift.
Step 2
Substitute and calculate
The argument is 1.08−(-2)=3.08. Change of base gives ln(3.08)/ln(4)=0.811465; raising 4 to this output recovers 3.08.
Step 3
Check the mathematical meaning
The domain is x>-2, and 1.08>-2. The inverse check is 4^(0.811465)≈3.08; adding -2 recovers x=1.08. Allowed input x: 1.08; Logarithm value: 0.811; Base raised to output: 3.08. Decimal values are rounded, so use unrounded intermediate values.
The answer
The argument is 1.08−(-2)=3.08. Change of base gives ln(3.08)/ln(4)=0.811465; raising 4 to this output recovers 3.08. The domain is x>-2, and 1.08>-2. The inverse check is 4^(0.811465)≈3.08; adding -2 recovers x=1.08. Animation check: Allowed input x: 1.08; Logarithm value: 0.811; Base raised to output: 3.08. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
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The mathematical idea
A real logarithm requires a positive argument. The shift a moves that boundary, while a base greater than one controls how quickly output grows. Exponentiating the output recovers the positive argument, not the original x until the shift is added back. Let f(x)=log_4(x−(-2)). The base is 4>1. Use x=1.08; the logarithm argument is exactly 3.08. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
y=log_b(x−a); bʸ=x−a; x>a
03 · Reflect and transfer
Explain what changes and why.
With the argument fixed, why does increasing a base greater than one decrease this positive logarithm?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.