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We consider a predictor–corrector numerical method for solving Caputo–Hadamard fractional differential equation over the uniform mesh <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>log</mi><msub><mi>t</mi><mi>j</mi></msub><mo>=</mo><mi>log</mi><mi>a</mi><mo>+</mo><mfenced separators="" open="(" close=")"><mi>log</mi><mfrac><msub><mi>t</mi><mi>N</mi></msub><mi>a</mi></mfrac></mfenced><mfenced open="(" close=")"><mfrac><mi>j</mi><mi>N</mi></mfrac></mfenced><mo>,</mo><mspace width="0.166667em"></mspace><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>N</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>log</mi><mi>a</mi><mo>=</mo><mi>log</mi><msub><mi>t</mi><mn>0</mn></msub><mo><</mo><mi>log</mi><msub><mi>t</mi><mn>1</mn></msub><mo><</mo><mo>…</mo><mo><</mo><mi>log</mi><msub><mi>t</mi><mi>N</mi></msub><mo>=</mo><mi>log</mi><mi>T</mi></mrow></semantics></math></inline-formula> is a partition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><mi>log</mi><mi>a</mi><mo>,</mo><mi>log</mi><mi>T</mi><mo>]</mo></mrow></semantics></math></inline-formula>. The error estimates under the different smoothness properties of the solution <i>y</i> and the nonlinear function <i>f</i> are studied. Numerical examples are given to verify that the numerical results are consistent with the theoretical results.